A Sobering Thought: There May Be No Credible Biblical Accounts Of Jesus' Final Days

                                                                       

 According to Yale Religious Studies Prof. Dale B. Martin it is more than probable that there never was a trial of Jesus at all. As he puts writes in his Yale University monograph (Introduction to New Testament History and Literature, p. 181 )

 "The Romans did not need to try a troublesome Jewish peasant in order to kill him."  

 In other words the entire trial tableaux -  while not manufactured out of whole cloth - was clearly constructed by later copyists, scriptural writers who desired a particular representation.  This would be to aggressively promote belief in a Messiah.

As Prof. Martin relates (ibid.), the Romans stoned or crucified troublesome, nameless lower class people all the time.  In the case of Jesus, who "had caused a disturbance in the Temple and made radical pronouncements"  the Romans would have "simply taken him by force and crucified him the next day" with a few other nameless  others.  Adding:  

"There was no need for any trial much less two or more different 'courts'.  It would have been more trouble than a Roman governor needed for the desired result."

"Moreover even if there were any kind of trial there is no way any information about it would have been transmitted to his disciples so they could then pass the stories along so they could be recorded in the gospels." 

Further there is no way any lower class peasant follower - say Peter- could have gained access to any actual hearings. Indeed, more than 30 years separated the events at the time from the first recorded scriptures associated with the particular gospels.  The end effect is that all the related accounts, irrespective of which quadriform gospel - were confections, or creations. Myths if you will  or "fiction" short stories in today's parlance.  Or in the words of Prof. Martin (p. 184):

 "Any narrative of any trial is purely the product of  later Christian imagination which thought that since Jesus was the most important man in history there must have been significant trials before his execution."

Prof. Martin is careful on the next page, however, to emphasize that this "historical Jesus" is in no way dependent on the Christian faith. So one can still believe (as a Christian) and have faith in Jesus but ignore the elaborate biblical mythology erected around him. One can also choose to believe these stories, but ought to know or realize there is no exegetical (or historical) support for them.

All of which conforms with the conclusion of the Catholic Scriptural historian, the Rev. Thomas Bokenkotter, who wrote in his monograph (A Concise History of the Catholic Church), p. 17:

"The Gospels were not meant to be a historical or biographical account of Jesus. They were written to convert unbelievers to faith in Jesus as the Messiah, or God.”

One central problem for the conventional Christian believer inevitably arises in all these exegetical discussions:  how to reconcile his/her faith in a 'God-Man/Savior' Jesus, with the actual historical person. This historical person was more a radical, "liberal" freedom-fighter against the Roman state than a God-man.

John Dominic Crossan in his remarkable monograph :The Historical Jesus: The Life of a Mediterranean Jewish Peasant, offers a hint ('Epilogue', p. 423):

"Is an understanding of the historical Jesus of any permanent relevance to Christianity itself? I propose that at the heart of any Christianity there is always, covertly or overtly, a dialectic between a historically read Jesus and a theologically read Christ. Christiany is always, in other words, a Jesus/Christ/ianity."

And finally (ibid.)

"This book challenges the reader on the level of formal method, material investment, and historical interpretation. It presumes there will always be divergent historical Jesuses, that there will always be divergent Christs built upon them, but above all, it argues that the structure of a Christianity will always be: *this is how we see Jesus as Christ now*." 

Words to bear in mind today, and over this weekend.

See Also:

• A Critical Biblical Hermeneutics Approach To the Easter Story...

And:

• A Brief History Of Early Christianity

And:

• Examining the Basis Of Biblical Hermeneutics. (Pt. 1)
• 
  

And:

• Examining the Basis Of Biblical Hermeneutics. (Pt. 2)
• And:
• • Religion

    Inspection- Was Jesus Gnostic?

    by Ken Carman | March 31, 2024 - 2:05pm | permalink

    

  And:

  Donald Trump | Religion

  Trump Bibles Make a Mockery of Christianity — And That’s Exactly Why MAGA Will Eat Them Up

  by Amanda Marcotte | March 29, 2024 - 6:32am | permalink

  — from Salon

  

• 
  

Savage DOGE Cuts Hit Astronomy- Space Science Research & They May Well Be Beyond The Point of Recovery

 

Solar physicists brainstorm at the Helio Hive workshop in February.

After DOGE's savage cuts there may never be another.

"So far, we have treated the various assaults of President Trump and the acolytes in his administration as a series of different attacks. In one lane they are going after law firms. In another they savaged U.S.A.I.D. In another they’re attacking our universities. On yet another front they’re undermining NATO and on another they’re upending global trade.

But that’s the wrong way to think about it. These are not separate battles. This is a single effort to undo the parts of the civilizational order that might restrain Trump’s acquisition of power. And it will take a concerted response to beat it back."-  David Brooks, NY Times, What’s Happening Is Not Normal

Two months ago, the critical role of the Solar Dynamics Observatory was featured at the 'Gathering of the Helio Hive' we attended in Boulder, e.g.

SDO 2025 Science Workshop: A Gathering of the Helio-hive!

At the time, at our first small group breakfast at the Boulderado Hotel - where the workshop was held - organizer Dean Presnell warned that Musk's DOGE fanatics could devastate NOAA. He predicted that up to ten percent of those giving presentations could "find themselves on breadlines".  In the wake of that warning we worried about just how much astronomy, astrophysics and planetary science could be savaged if the DOGE monster struck federal grants from NASA as well as other agencies.

The answer was not long in arriving, in an email 3 days ago from Gordon Emslie: Chair of the Solar Physics Division Public Policy Committee.  He wrote:

You may have seen this recent article outlining President Trump’s proposed FY26 budget proposal that would decimate NASA Science.

Under this proposal, NASA’s Science Mission Directorate would receive almost a 50% reduction in its Budget. Within the individual SMD Divisions,

* The Planetary Science budget would decrease from its current level of $2,717 Million to $1,929 Million;

* The Earth Science budget would decrease from its current level of $2,195 Million to $1,033 Million;

* The Astrophysics budget would decrease from its current level of $1,530 Million to $487 Million;
and

* The Heliophysics budget would decrease from its current level of $805 M to $455 M.

The AAS has issued a Press Release expressing “grave concern” over these proposed budget levels which, if enacted, would have an absolutely devastating impact on our community and on our nation’s leadership in science. However, Congress has the final word on federal funding.

The last bullet on Heliophysics cuts was especially devastating to learn for those of us in the solar physics community (of which I am an Emeritus member).  This means the Solar Dynamics Observatory itself could soon have funding pullled, truncating its continued access. This is especially tragic given that "hundreds more predictive parameters" had been made available in the last 20 years.  This according to the guest lecturer at the first plenary session of the Hive workshop. 

If this observatory is scuttled then, with all those predictive parameters no longer accessible, how will we be able to predict the most violent space weather events including potentially Earth-exterminating superflares, e.g.

Brane Space: Solar Superflare Potential Now Accepted As More Common Than Once Believed

Not to mention fewer insights into the dynamics of the solar corona, e.g.

Solar Rollercoaster: looping-the-loop in the solar corona

Emslie in his email communication then advised we contact representatives to "Help ensure these proposed cuts do not come to fruition by urging your Representatives to sign onto the House Letter to Appropriators that requests $9 Billion for NASA SMD in FY 2026."

But of course the House is currently under the control of Trump bootlickers, apparatchiks and sycophants, so how far will any such appeal get in Mike Johnson's House? Well, not very far at all - at least not until the Reepos find their spines.

Obviously, it'd be a veritable godsend if this could work. As Emslie notes:

"The level of funding requested in the bipartisan letter will ensure that all NASA Science Areas can thrive – from robust flagship missions and a steady cadence of new Explorers, to low-cost access to space opportunities, technology development, and supporting research. Overall, it would enable a strong workforce to deliver the groundbreaking science that advances the next generation of science and exploration as outlined in the community decadal survey reports."

But until and unless the Trump cult Reeps are ejected from the House (Next year's midterms) that aspiration will remain a pipedream - not only for space science but for all science.

See Also:

Big Data | Donald Trump | Elon Musk

The Department of Government Efficiency Is a Lie — And It Might Destroy Us All

by Thom Hartmann | April 17, 2025 - 5:12am | permalink

— from The Hartmann Report

America is under siege, but not by foreign armies or rogue terrorists. Instead, the threat is coming from inside the house: unelected billionaires who are quietly hijacking our democratic institutions from within.

Although it’s entirely possible that they are collaborating with foreign autocrats, including Vladimir Putin, who are also billionaires and thus share their unique authoritarian worldview.

And thanks to the bravery of a whistleblower named Daniel Berulis, we now have a window into what may well be one of the most brazen and dangerous power grabs in American history.

This isn’t cheap or even ordinary corruption: this is the sort of thing that happens in Third World countries, like when Ferdinand Marcos in the Philippines used cronies and spies to infiltrate government agencies and neutralize political opponents, or when Vladimir Putin installed former KGB allies into key Russian institutions to crush whistleblowers and rig judicial outcomes, or when Guatemala’s President Jimmy Morales dismantled anti-corruption agencies and handed sensitive state information to business elites.

» article continues...

And:

• The Ars Technica article on the proposed NASA FY 2026 Budget: https://arstechnica.com/space/2025/04/trump-white-house-budget-proposal-eviscerates-science-funding-at-nasa/

• The AAS Press Release expressing concerns about the proposed budget: https://aas.org/press/aas-statement-nasa-cuts

Some Basic Elements Of Abstract Algebra - Part Two (Groups)

Groups, as noted in my April 11 post, comprise the 2nd large subset area of what we call modern or abstract algebra. We now launch into some of the basic elements and aspects:

 1. Definitions:

A GROUP is a set G with one binary operation defined on it , and G satisfies the axioms:

(a) Associative law: a· (b · c) = (a· b) · c

(b) Identity element (e): there exists an element e  Î  G such that: e · a = a for all a Î G and a · e = a

(c) Inverse element: For any a Î G there exists an element a ^{- 1}  Î G such that:
a · a ^{- 1} = a ^{- 1} · a = e

d) Commutativity[1]: Only if there exist elements a, b Î G such that (a · b) = (b · a), then G is said to be an Abelian Group.

2. Generating Simple Groups:

A) Clock Groups:

These are amongst the simplest of all, and easiest to generate – they also represent a nice introduction to modular arithmetic, the Euclidean Algorithm etc.

Let a clock group representation be denoted as shown in the diagram of Figure 1.

We seek to construct a group from the above, which is closed under addition. As we can see there are four members, 0, 1, 2, and 3. The process of addition is defined by adding elements – starting with 0- in a clockwise sense. Doing this we should be able to find a complete closed set of addition operations for all the elements. For example, we find 0 + 1 =1, and 0 + 2 = 2 and so forth. Similarly, we find 1 + 1 = 2, 1 + 3 = 0, 2 + 3 = 1 and so on. Each result obtained by adding the portion of the cycle from the starting element. From here, we may set out the group under addition (+) (Fig. 1- right top)

The reader should easily be able to check each of these and demonstrate for himself that the table is valid. Is G+⊕ (4) a group? Yes, because it obeys all the properties for a group.

Problem: For the same group, develop a table to show it is closed under multiplication (x). Hint: Simply extend the principle of addition to the case of multiples, and start all multiple entries counting clockwise from 0. For example, 2 + 2 + 2 amounts to three two’s counted from 0. (Three sets of two). One such counting set leads to 2, and two leads to 0 and three leads to…? Obviously, 2.

Thus: 4 + 4 = 0 and 4 + 4 + 4 = 0 so that 3 x 4 = 0

What about: 3 + 3 + 3?

By inspection and using the clock graphic we obtain: 3 + 3 + 3 = 1 so that 3 x 3 = 1

We thereby arrive at the (x) table for the group G+⊕ (4) (Bottom right in Fig. 1)

Is this group commutative? Some checks of the operation using pairs of elements will confirm that it is (E.g. 2 x 3 = 3 x 2). Hence, we can aver it is an Abelian group.

Problem: Set out a clock group with five elements (G+⊕ (5)), and prepare tables to show the group is closed under (+) and (x).

Solution:

The correct diagram is shown in Fig. 2, along with the tables for multiplication and addition.


B) Cyclic Groups and Sub-groups

A more advanced variation on the simple clock group is what's called a "cyclic group" which we will see can also be a sub-group. To clarify definitions here - let G denote a group, and let H be a subset of G then H is a subgroup of G if: H is closed under the same operation as G, for each element x there exists the inverse element, x ^{- 1}, and these are related to an identity element I such that: (x)(x ^{- 1}) = I.


A relevant theorem- Lagrange's Theorem:

Let G be a group of finite order n, then let a be any element of G. Then the order of the element divides the order of G.

Proof:

Let H be the set of all powers of a:

H = {a, a² , a³ , a⁴...........a ⁿ}

But, H must have a finite number of distinct elements.

Then: a ⁿ = a ^m (for some m)

a ⁿ(a ^m) ^{- 1} = a ^{n - m}   a ^m (a ^m) ^{- 1} = a ^{n - m} *I

Is the cyclic group closed under (x)? Check by consideration of C ₄, the cyclic group of order 4. To see an example of C ₄ simply take the diagram for the clock group, G+⊕ (4), and re-assign it elements as follows:

0  ® 1
 
1  ® a ¹ = a

2 ®  a ²

3  ® a ³


Now, what will the multiplication table look like?

Solution: the result is shown in Fig. 3.  Note especially how the table differs from the (x) table for the clock group, G+⊕ (4).


Practice exercise:

Using the diagram for the clock group G+⊕ (5) in Fig. 2 as a template, work out the elements for the cyclic group C ₅. Prepare a table for (x) applicable to its elements. Is  C ₄  a sub-group of  C ₅? Why or why not?
--------------------

[1] This is not a critical, or indispensable group property. If it does apply, we say the group is Abelian

Bill Maher Dotard's "Friend"? - In Fact More Like A Useful Idiot After His Gushing 'Real Time' Episode

   Maher glorifies Trump for 13 minutes on his Friday show - proving even smarties can be played

"The world is now seeing Trump’s America for exactly what it is becoming: a rogue state led by an impulsive strongman disconnected from the rule of law and other constitutional American principles and values. And do you know what our democratic allies do with rogue states? Let’s connect some dots.

First, they don’t buy Treasury bills as much as they used to. So America has to offer them higher rates of interest to do so — which will ripple through our entire economy, from car payments to home mortgages to the cost of servicing our national debt at the expense of everything else." - Thomas Friedman,  NY Times, "I have Never Been More Afriad For My Country's Future'

Bill Maher in a nausea -inducing 13 minute romp on Real Time Friday night, gushed that on a recently invited (by Kid Rock) WH visit, he found the president “gracious and measured and able to laugh at himself."  Cue the barf bags. Of course he was able to laugh at himself after he pulled the freakin' wool over more than 77 million voters on November 5th, and got the morons to interject a 34 count indicted felon back into power. 

A power governed by retribution even as hundreds of Venezuelans find themselves in a Salvadoran hellhole prison with no chance of escape. Oh, and college students find they can't get access to financial assistance because Dotard blew up the Dept. of Education.  Oh, and we now learn NASA's top astrogeology section is being dismembered under DOGE.

Sorry, but any asshole that sucks up to a guy who's responsible for all that is a useful idiot.  And being played by an orange traitor clown who likely can't believe such a PR stunt and optics win fell into his lap.  Golly freakin' gee he ain't nutso after all, he's "measured".  

How much of a loser asshole do you have to be to get "charmed" by the Dotard? Well, one of cosmic scale for sure, proving even an erstwhile smartypants like Maher can be suckered into the orange toad's fungal atmosphere and not know it.  Don't believe me?  Describing their hours-long meeting and meal on March 31, Maher chirped that Trump gave him “a very generous amount of time, and a willingness to listen and accept me as a possible friend."  

Accept you as a possible friend? You mean another ass kisser like the dozens of butt kissing Reepo sycophants in congress? Who won't lift a damned finger to put a check on Trump's lawless overreach? Sorry, boyo, you've now been relegated to the segment of Trumpers who fawn over a lawless, power mad renegade and ensure - by your 13 minute slavering- even more power and influence as he craters the global economy.

Fortunately, one WaPo columnist and author (Chaos Under Heaven) - Josh Rogin - was present to puncture Maher's fantasizing. After Maher had gushed about the Dotard being "warm and measured", Rogin pushed back and informed the wide-eyed imp he'd been used as part of a PR stunt. After all, to get a popular comic with an influential show to call a madman warm and measured has to be a PR (and propaganda) win. As Mr. Rogin put it:  

"I think you've fallen into the trap, and I think I represent 99% of the internet when I say this: you've played the game of proximity is principle. I'm not questioning your motivation, I'm questioning Trump's, OK?"

Hard to contradict that, certainly from what I saw Friday night.  Wondering to myself if Maher had smoked one too many reefers before his WH visit. 

 Certainly when Maher went off the rails when he asserted Dotard was "one of the most effective politicians because of his authenticity"  i.e.   

Bill Maher Calls Trump One Of The 'Most Effective Politicians' Because Of His Authenticity

One had to question if Maher too had lost all his marbles. Was he even in the same universe assessing an orange porker who'd already reeled off more than 37,000 lies?

Of course, Maher took umbrage that he could be so questioned for going to an hours long visit with the worst predator president ever, sending this nation over a cliff piece by piece. Confirming he's doing Putin's dirty work for him, and once again repeating the lie that Ukraine started the war with Russia. yeah right, and monkeys type out works of Shakespeare on their infinite typewriters.  Anyway, after shutting down Rogin, Maher barked:

"Did you hear what I said? What is the alternative to not talking? Just sit at your lunch table and don't talk to anybody?"

Which shows he's as clueless as he is naive.  Because Rogin never said don't talk to anybody. If there is a Trumper not totally sucked into Trump's butthole and open to sane, logical discussion, by all means. But that is not the same as being sucked into the orbit of a traitor and felon and allowing his pseudo-charm to turn your brain into a propaganda machine.

As one recent Washington Post column put it:

"The remarks amounted to a win for the White House: One of the president’s most prominent critics, for a night, spent 13 minutes on a monologue humanizing Trump, rather than caricaturing or attacking him. Elon Musk, Trump’s billionaire ally, shared a video of Maher’s remarks that had received more than 55 million views as of Monday morning."

For the rest of us, I warrant most of the sane segment of this country, Maher's glib glorification of a traitor - who pardoned thousands of January 6th traitors - is an atrocity. One we won't soon forget.  Also, that Maher himself has lost all contact with reality, especially when he yapped:

"A crazy person doesn’t live in the White House. A person who plays a crazy person on TV a lot lives there,”

Sorry, Billy boy, but if you don't recognize you've made a sociopath and fascist sound acceptable, then I can't help you.

Some WaPo comments on Maher's meet:

-Maher should have just administered the BJ while he was there in the room. Trump earned it for the time he spent snowing this guy.

I saw the show and was appalled. I don't care how funny Trump is, or even if he has a sense of humor. For all the horrible things Trump has done to his own country and the world, being funny is the least important thing..

There are two possibilities: Maher is gullible and not that bright, so he got snookered. Or, Maher is verging toward Trumpism and not that bright, so he allowed himself to get snookered.

Like Trump it would seem Maher has had a "He likes me!" moment. I never would have believed he could be that naive. You've joined the multitude of folks who have been hoodwinked by the creatures 'charm' offensive, Bill. Sad.

I look forward to when Maher gets a tour of the Salvadoran prison  like his delightful tour of the WH and declares as a smug “centrist” his equivalent of “there’s nice people on both sides.”

Maher’s comments only heightened my concern about the country’s future. When I thought the Orange One was just an amoral jerk selling himself to right wing oligarchs, life was just about survival until 2029.The portrait Maher paints of a sane rational being creating a caricature to promote chaos is scary.

See Also: 

Donald Trump | Fascism

‘Don’t You See What’s Happening?’ Experts Call Trump’s Latest Moves ‘Full Blown Fascism’

by David Badash | April 15, 2025 - 6:20am | permalink

— from Alternet

President Donald Trump’s recent escalating rhetoric and actions across multiple fronts have alarmed political experts, who are now warning that the United States is not just drifting but accelerating toward fascism—and may have already crossed the threshold.

“He’s threatening media companies who are critical of him,” warned Republican Sarah Longwell, a political strategist and publisher of The Bulwark. “He’s talking about sending Americans to foreign prisons. He’s signing executive orders to investigate former staff members who spoke out against him. Don’t you see what’s happening here?”

Trump on Sunday night attacked CBS and its “60 Minutes” newsmagazine, a top-rated and esteemed broadcast for more than five decades. The President, apparently angered by its reporting, called for CBS’s broadcast license to be revoked. He also called on his hand-picked head of the Federal Communications Commission (FCC) to “impose the maximum fines and punishment, which is substantial, for their unlawful and illegal behavior,” namely, reporting facts he did not like.

» article continues...

And:

Donald Trump

A Couple of Cool Dictators Sittin' Around Talking

by Jaime O’Neill | April 16, 2025 - 5:10am | permalink

You heard that right, right? Donald Trump, the current President of the United States, a country once thought to be against dictatorial rule and generally inclined to favor democracy and the rule of law, praised the ruler of El Salvador as one of the coolest dictators currently dictating. He also showed yet more inconsistency of how to dress when coming to the Oval Office. Not cool to come not wearing a tie if you're the leader of Ukraine. You will be scolded and bullied. On the other hand, Kid Rock is so cool he can come wearing a Halloween costume. Same deal with Bukele of El Salvador who showed up tieless only to be praised lavishly.

There are quite a few dictators in the world, btw and fyi, though none of them are anywhere near as cool as Trump, himself, of course, but that went without saying. When it comes to being a dictator, nobody is currently doing it better, and nobody is cooler than DJT. This Bukele dude in El Salvador would be wise not to let that praise from the master go to his head.

» article continues...

And:

Donald Trump

Chaos

by Robert Reich | April 14, 2025 - 5:24am | permalink

— from Robert Reich's Substack

As tens of millions of Americans hussle to pay their taxes, Trump has put the entire global economy into chaos. 401(k)s are tanking, savings are shrinking, treasury bonds are losing value, supply chains are convulsing.

Even America’s oligarchs are petrified. They contributed millions to Trump’s inauguration. Many invested heavily in his campaign. They lavished praise on the new president and have supported his every move — in order to benefit from his promised big tax cut.

But the chaos he’s unleashed on the world economy is causing many of them to go public with their worries.

“Obviously,” Jamie Dimon, JPMorgan Chase’s chief executive, said in a conference call with reporters, “the China stuff is significant. We don’t know the full effect.”

» article continues...

And:

There Is No Choice: Trump Must Be Impeached

Excerpt:

Trump has figured out how to convert the slow grind of due process into a tool of mockery. Gag orders become proof of censorship. Trials are dismissed as witch hunts. The very idea of law is hollowed out and rebranded as tyranny. And the longer the system hesitates, the deeper the damage. Rule of law begins to resemble a punchline.

We’ve seen this before, just not here. From Orbán to Erdoğan to Putin, we’ve watched elected leaders keep the outward forms of democracy – elections, courts, legislatures – while draining them of substance. The systems look intact, but their spirit is gone. Power is centralized, opposition is marginalized, and the rule of law becomes rule by law. Trumpism is not some American anomaly – it is part of a broader global drift toward strongman politics cloaked in democratic rituals…

 That’s the grim lesson for us now: there is no clean exit from this crisis, no perfect choice. But doing nothing is not caution – it’s capitulation. A constitutional democracy cannot survive if it’s afraid to enforce its own rules.  If this moment doesn’t justify impeachment, then the mechanism is a dead letter. 

And:

Bill Maher went to Washington. He got played.

Excerpt:

Trump made him feel comfortable and listened to. In their conversation, which included “an amazing tour of the whole house,” Trump remained empathetic. “He didn’t get mad. He’s much more self-aware than he lets on in public,” Maher recalled. “Why can’t we get the guy I met to be the public guy?” 

It’s unfortunate that Maher allowed himself to get played like this. In private, many of history’s most brutal leaders have been described by their contemporaries in surprisingly human, even affectionate, terms.

By agreeing to meet with Trump, allowing himself to be privately charmed by a charismatic leader and then sharing his softened new take on the president, Bill Maher has made the task of holding Trump accountable that much more difficult. 

Solutions To Basic Elements Of Abstract Algebra Problems

  Solutions: 

(1) By the Euclidean (division) algorithm there exist numbers q, r ∈  Z

Such that 0 < r < n,   and a is of the form:

a = q n + r

By definition of   [a ]  mod n  =   [a]

 r  ∈   [a] ,   And:   r = a + (-q) n

  (2) Let Z be the integers.  The ideal:

 

           (5)  =   {5 j:  j ∈  Z }

Show all the congruence classes with respect to this ideal.    

(Hint:  [a] =  {a + j: j ∈ [(5)}  = {a +  5j: j ∈   Z } )

The congruence classes are:

[a]   =  {a + j: j  ∈  (5)}  =  {a + 5j: j ∈  Z }

[0]  =  {0 + 5 j: j  ∈  Z}  =  { 5j: j ∈  Z }  =  (5)

[1] =  {1  + 5 j: j  ∈  Z}   =  {1, 6, -4, 11, -9}

[2]  =  {2  + 5 j: j  ∈  Z}   =  {2, 7, -3, 12, -8}

[3]  =  {3  + 5 j: j  ∈  Z}   =  {3, 8, -2, 13, -7}

[4]  =  {4  + 5 j: j  ∈  Z}   =  {4, 9, -1, 14, -6}

[5]  =  {5  + 5j: : j  ∈  Z}   =   5 (j+ 1) : j  ∈  Z}   =  [5] = [0]

3) Using set notation we may define:

A ·  B  =  {x ·  y: x ∈ A,  y ∈ B}

A +  B  =  =  {x +  y:  x ∈ A,  y ∈ B}

4) Take S as the set of integers, Z. Let the ideal I = (2) so that S / I =  Z ₂     Thence or otherwise, find:

a (a)     [0]     b)   [1]     c) S/ I  =  Z ₅       

 

Take S =    Z.  I = (2)  so that S / I =  Z ₂    

a) [0] =  {0, + 2, + 4, + 6…..}  =  I = (2) 

b) [1]  =   {1, 3, -1, 5, -3, 7….}

c)If S / I  =  Z ₅       

S/I =  {[0], [1], [2], [3], [4]}

5 ) Show every ring S has two ideals: S itself and {0}.

Every ring S always has at least two subrings, namely S and the zero ring,  S _0.

Further, if S is also a  field then the only ideals in S are S and {0}.  Further:

 if 1   ∈ I  then I = S. (Let I = S be an ideal,  I = {0} is also an ideal. )

 (In a field F, the only ideals are 0 and F)

Basically, an ideal of a ring S is an additive subgroup a  of  S with the property that:

a   ∈ S    and a  ∈  a    imply  ra  ∈  a  .  Clearly then the set containing the single element 0 and the set containing the whole ring S are ideals.