Tuesday, December 9, 2025

Basic Elements Of Abstract Algebra Solutions (Part 3)

1)  Find all ideals I of the integer class Z 12    and in each case compute:

 

Z 12  / I  

 

I.e. find a known ring to which the quotient ring is isomorphic

 

Solutions:


=  [0] ,   Z 12  / I    ~    Z 12  


=  {0, 2, 4, 6, 8, 10;  Z 12  / I 2 }  ~    Z 2  

 

=  {0, 3, 6, 9; 10,  Z 12  / I 3 }  ~   Z 3  

 

=  {0, 4, 8;   Z 12  / I 4 }  ~   Z 4  

 

=  {0, 6;   Z 12  / I 5 }  ~   Z 5  

 

=  Z 12 ,  Z 12 / Z 12 }  ~    0 

 

2) Find a subring of the ring Z + Z  which is not an ideal of Z + Z

 

Solution:

 

{n, n   |    Î Z}


3) Give all units in each of the following rings: 


 a) Z   b) Z + Z   c)  5    d) Q


Solutions:  

 a) 1, -1   

 b) (1,1), (1, -1), (-1, 1), (-1, -1)   

c) 1, 2, 3, 4  

d) all non zero q  Î Q  


4) Let S be a commutative ring with I an ideal a  Î S, with (a) = {ja : j Î S},  

Prove that S is closed under +   and ·   

Soln. 

We assume x Î (a),   y Î S,   

Then x = ja for some j Î S,   So:  · y(ja)=  (yj) a Î a  And we can write the following formulation:  If x,y  Î (a) then x + y  Î (a),  

 Also: x = ja for some j Î S,  then x - y   Î (a)  and further, y = j' a for some j' Î S then x· y  Î (a)   

 Then for (+) closure: x + y = ja + j'a = (j + j') a  Î (a)   

 For (· ) closure:  x · ja  ·  j'a  =  (j - j') a Î (a)  

And for subtraction: x - y =   ja - j'a = (j - j') a Î (a)     

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

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

    Solns.

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

        We have:

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]}


Monday, December 8, 2025

Trump's Pardon Of Honduran Narco- Kingpin Proves His Attacks On Fishing Boats Have Squat To Do With Stopping Drugs

                                                                    
             "Yeah, I'm gonna pardon El Chapo's pal, Juan Hernandez! So sue me"
 
Narco-terrorist Juan Hernandez dumped 400 tons of cocaine into U.S.

Fishing boat allegedly ferrying drugs drugs about to be smoked


"Blowing shipwrecked sailors out of the water was the sort of thing the Japanese and Nazis did during World War 2.  The four star (Bradley) should be court-martialed and charged with a war crime along with Hegseth. The pardoning of the former president of Honduras should prove these actions are not about drugs." Denver Post Letter to the Editor yesterday.


Traitor Trump bellowed Saturday about possibly launching land attacks in Venezuela and beyond to "Stop drug cartels" threatening the U.S. (sic). But by now  it's clear this is all bullshit. The tell?  Pardoning one of the worst narco-traitor cocaine pushers in the Western Hemisphere. Because, logically, if he was genuinely serious in his claims to "protect the U.S. from drug cartels",  he'd never have given a get out of jail card to a guy that dispatched over 400 tons of coke into the U.S. Nor would he be approving mass murder on the high seas, international waters no less - including killing survivors in an already blown up boat.

As I've written before, the rules governing conduct in a presumed war theater are very well defined by the Geneva Convention. To wit, you cannot launch indiscriminate attacks on alleged enemies without any evidence, and you certainly cannot slaughter survivors in cold blood. 

'To date the infernal and impious felon president has done both (killing up to eighty-six people), even while having pardoned one of the worst narco- trafficking  terrorists in U.S, history.  Which act makes a total mockery of his claim he is ordering small boats be blasted to bits to "protect Americans from fentanyl" and other drugs. (MJ, coke).

The idiot doesn't even know, certainly isn't aware - that Caribbean drug runners - if that is what they are- do not convey fentanyl. That is out of their league.  At most they may be carrying cocaine, but that is irrelevant if Trump pardons a cocaine "king" who had 400 tons of the stuff dispatched into the U.S. in one go.

Who is this real narco terrorist  Trump pardoned? According to The Wall Street Journal ('Ex-Honduran Leader Trump Pardoned Ran Violent Narco State',  December 2, p. A8) the pardon of former Honduran president Juan Orlando Hernandez

 "was a dizzying reversal of fortune for a Central American leader who U.S. prosecutors said enabled drug bosses to smuggle hundreds of tons of cocaine into the U.S. and turned his country into a violent narco-state. He was sentenced to a 45-year prison sentence."

Process that! This is the maggot Trump pardons - PARDONS! - while he blasts alleged small drug carriers (misnamed "narco terrorists")  to bits in the Caribbean, without even having provided one scintilla of evidence that: a) they were actually ferrying drugs, and b) they were any more than Caribbean fisherman trying to make a few extra bucks to feed families.

As one writer in another section of the press has put it: 

"The Trump administration has been blowing up fishing boats in the Caribbean — and now one in the Pacific — claiming without evidence that they’re “drug boats.”

These are extrajudicial executions outside any system of law. And there’s a reason we shouldn’t allow drug warriors to act as judge, jury, and executioner: because over the years, they’ve made many, many tragic mistakes and killed lots of civilians.

What makes these strikes so appealing to Trump is that it gives him the godlike power to look down from above and smite anyone who displeases him, without consequence. He’s even told sick jokes about local fishermen in the Caribbean now being afraid to get in their boats."

 That was written over a month before Trump's pardon of the most notorious drug henchman in Central America. A thug who'd been given 45 years and Trump let out after serving barely one.  Like he pardoned the nearly 1,600 traitors who attempted to overthrow and torch the Capitol.  What is going on? Easy, we have a traitor and felon in power who is willing to release (or interfere in) the case of any fellow felon or traitor (like Jaire Bolsonaro in Brazil) who he believed "got a raw deal'.

As the WSJ piece expatiates:

"In power from 2014 until his arrest days after leaving office in 2022, Hernandez used his political influence to allow smugglers to ship more than  400 tons of cocaine into the U.S.  In exchange  for protecting drug loads he received millions of dollars from some of the world's most powerful and violent cartels, money he used to fuel his political rise.

Hernandez's younger brother was among the drug bosses who prosecutors said profited from the former president's connections. During the younger Hernandez trial a witness testified that Mexican drug boss 'El Chapo' Guzman gave $1 million in cash during a 2013 meeting to help Juan Orlando Hernndez's presidential campaign. The expectation was that Guzman's cocaine routes would be protected after Hernandez became president."

"Prosecutors described Honduras as a state where drug traffickers aligned with Hernandez's National Party were completely protected from investigation and arrest. The cartels even sent troops to the Honduran border with Guatemala to protect their turf."

Adding at the end:

 "Trump's planned pardon shocked many observers, including Republicans. The pardon comes as the Trump administration is blowing up small boats, allegedly carrying drugs in the Caribbean and Pacific. 'Why would we pardon this guy and then go after Venezuelan president Nicholas Maduro for running drugs into the United States ' wrote Sen. Bill Cassidy (R, La) on X."

Elementary, Mr. Cassidy. Because a convicted felon like Trump will always extend favor to another felon, and damned adherence to the law or justice.  This was before the foul drunk maniac Hegseth spouted at a Pentagon presser on Saturday - about further boat strikes in the Caribbean:

"And we'll keep killing them until they stop poisoning our people!"

The drunken Turd clearly oblivious to his master Trump pardoning a guy who enabled cocaine shipments that likely poisoned tens of thousands of Americans. Sorry, Bubba, two systems of justice - one minus any evidence and carried out without due process - ain't worth lickspittle.  And yet a White House spokesperson (also on Saturday)  having the absolute gall to spout this further hogwash:

"The narco terrorist shipments in the region directly threaten Americans and the national security interests of the United States."

Sorry, you don't get to use a double standard when you have a so-called "president" who just pardoned a genuine narco terrorist who did a thousandfold worse, and got to WALK!  Clearly the actions of this orange felon mean the national security prattle is pure smokescreen and merely used as an excuse to commit extra-judicial murder as the orange traitor decides.

This is also why the Armed Services Committee needs to not only go after Hegseth for ordering the murder of survivors in the Sept. 2 re-strike (not "double tap") but also Trump himself - demanding why he pardoned a bloody narco thug while trying to overthrow Maduro. 

Even The Wall Street Journal in a recent editorial ('Shooting The Wounded On Drug Boats', p. A14, Dec. 2) is adamant that congress needs to act, i.e.

"Every time the Senate Armed Services Committee asks the Pentagon for more details they seem to get mostly a stonewall. That's all the more reason for Congress to learn the truth about the Hegseth story."

Yes indeed, we are ready to learn. And also why, up to now, the Trump cabal has not provided an iota of evidence that the boats it's blowing up with loss of life are transporting anything more than fish! In the meantime, we can assert that Trump's pardoning of the narco-trafficking felon Juan Hernandez shows he isn't serious at all about stopping drugs from entering the US of A. And hence, his "war"against the imagined "drug cartels" of Venezuela is bogus.

So if it's not about stopping drugs from getting into the U.S. what is it? In my reckoning it's plain old application of the old 'wag the dog' strategy. In this case start a fake war with Venezuela over "drugs" to keep attention of the media (and public) off the imminent release of the remaining Epstein files.

See Also:

Trump pardons major drug traffickers despite his anti-drug rhetoric - Anchorage Daily News

Excerpt:

On President Donald Trump’s first full day in office this year, he pardoned Silk Road founder Ross Ulbricht, who was convicted of creating the largest online black market for illegal drugs and other illicit goods of its time.

In the months since, he has granted clemency to others, including Chicago gang leader Larry Hoover and Baltimore drug kingpin Garnett Gilbert Smith. And last week, he pardoned former Honduran president Juan Orlando Hernández, who had been sentenced to 45 years in prison for running his country as a vast “narco-state” that helped to move at least 400 tons of cocaine into the United States

Overall, Trump - who campaigned against America’s worsening drug crisis and promised to crack down on the illegal flow of deadly drugs coming across the border - has pardoned or granted clemency to at least 10 people for drug-related crimes since the beginning of his second term, according to a Washington Post analysis. He also granted pardons or commutations to almost 90 others for drug-related crimes during the four years of his first term, the analysis showed.

And:

by Marjorie Cohn | December 6, 2025 - 6:12am | permalink

— from Truthout

Public outrage is mounting over the Trump administration’s September 2 “double tap” strike, in which the U.S. military bombed a small boat for a second time to kill the survivors of a first strike. This particular strike has garnered significant attention due to its clear violation of U.S. and international law because shipwrecked sailors should never be targeted. But it is crucial to note that Donald Trump’s entire bombing operation against vessels in the Caribbean and Eastern Pacific is illegal as well.

Trump’s campaign of extrajudicial violence under the pretext of fighting a “drug war” is reminiscent of the policies of former Philippine President Rodrigo Duterte, who is currently in custody in the International Criminal Court (ICC) in The Hague, Netherlands, awaiting trial for murdering alleged drug dealers and users. Like Duterte, Trump’s bombing campaign should be considered a crime against humanity.

» article continues...

And:

Opinion | The Laws of War Are Not Woke - The New York Times

Excerpt:

In their military campaign in South America, Donald Trump and Pete Hegseth aren’t just defying the Constitution and breaking the law. They are attacking the very character and identity of the American military.

To make this case, I have to begin in the most boring way possible — by quoting a legal manual. Bear with me.

Specifically, it’s the most recent edition of the Department of Defense Law of War Manual. Tucked away on page 1,088 are two sentences that illustrate the gravity of the crisis in the Pentagon: “The requirement to refuse to comply with orders to commit law of war violations applies to orders to perform conduct that is clearly illegal or orders that the subordinate knows, in fact, are illegal. For example, orders to fire upon the shipwrecked would be clearly illegal.”

Here’s another key line: “It is forbidden to declare that no quarter will be given.” A no quarter order is an order directing soldiers to kill every combatant, including prisoners, the sick and the wounded. The manual continues, “Moreover, it is also prohibited to conduct hostilities on the basis that there shall be no survivors, or to threaten the adversary with the denial of quarter.”

There are now good reasons to believe that the U.S. military, under the command of President Trump and Hegseth, his secretary of defense, has blatantly violated the laws of war. On Nov. 28, The Washington Post reported that Hegseth issued a verbal order to “kill everybody” the day that the United States launched its military campaign against suspected drug traffickers.

According to The Post, the first strike on the targeted speedboat left two people alive in the water. The commander of the operation then ordered a second strike to kill the shipwrecked survivors, apparently — according to The Post — “because they could theoretically call other traffickers to retrieve them and their cargo.” If that reporting is correct, then we have clear evidence of unequivocal war crimes — a no quarter order and a strike on the incapacitated crew of a burning boat.


And:

Yes, Hegseth IS A War Criminal - For Killing Survivors Of A Fishing Boat Attack With Most Occupants Already Slaughtered

And:

by Steven Harper | December 5, 2025 - 6:11am | permalink

On Friday, November 28, the Washington Post reported that when the US military began attacking small boats in the Caribbean, Defense Secretary Pete Hegseth had issued a verbal order that it should “kill everybody.”

When the first attack occurred on September 2, the initial missile strike destroyed the boat but left two survivors clinging to the wreckage. To comply with Hegseth’s order, Admiral Frank M. “Mitch” Bradley authorized a second strike that killed them.

If true, it’s a war crime.

Hegseth called the Post‘s reporting “fabricated, inflammatory, and derogatory,” claiming that the drug-smuggling operations were “lawful under both US and international law.”

» article continues...



Friday, December 5, 2025

Solutions To Mensa Algebra Squares Problem


We begin by dividing the positive integers into three groups:

(A) Odd positive integers, 1,3, 5, 7, 9 ....etc.

(B) Even positive integers of the form 4N, e.g. 4, 8, 12, 16...

(C) Even positive integers of the form 4N + 2, e.g. 2, 6, 10, 14, 18...

We note all positive integers (2N +1) CAN be written as a difference of two squares, i.e.

(2N + 1)  =  (N + 1) 2 -   N 2

We note all even positive integers  of the form 4N CAN ALSO be written as a difference of two squares, i.e.

(4N) =  (N + 1) 2 -   (N  -  1)  2


But all even positive integers of the form 4N + 2  cannot be written as a difference of two squares.

A difference of two squares is even when:

-  Both squares are even, but all even squares are multiples of four, so their difference is a multiple of four, e.g.

(2A) 2   -   (2B) 2  =  (4A) 2  -   (4B) 2   =  4( A 2   -  B 2 )


-   Both squares are odd, but all odd squares are one more than a multiple of four, so their difference is a multiple of four:

(2A + 1) 2   -   (2B + 1) 2  =  4A 2  + 4A + 1 -   4B 2  - 4B - 1

=  4( A 2  + A -  B 2 - B)





The Origin Of "End Times" Christian Theology And Why Israel Is Inextricably Bound To It

 

                                                                         
                         The book which started the evangelical belief that Israel would figure in Armageddon


We can safely say that Ralph Reed, Doyen of the Religious Right, doesn't understand jack about Evangelicals' obsession and adoration of Israel. Especially when he writes in his Nov. 7 WSJ column ('Evangelical Support for Israel Is About More than Theology', p. A14):

"A columnist for MSNBC warned that evangelicals believe 'war is not something to be avoided' but something inevitable, desired by God and celebratory'.

In this liberal fantasy, evangelicals embrace Israel because its existence begins the final countdown to Armageddon."

But before Ralph gets too cocksure about a "liberal fantasy" he ought to read Hal Lindsey's page turner: 'The Late, Great Planet Earth':

Lindsey’s 1988 effort directly tied Israel's birth in 1948 to the "biblical" prophecies for the rise of an "Antichrist" and Armageddon. Indeed, Israel's birth as a nation was essential to all of Lindsey's forecasts which have mesmerized millions of bible punchers.


Lindsey built his interpretation on the identification of Israel with the “fig tree” and the coming of "the Beast" within one generation of its emergence (birth of a Jewish state). The "Beast" is discussed in detail by Lindsey on page 103, and the link to biblical interpretation, i.e.

And then nailing it for his millions of bible puncher readers:

What grabbed readers was Lindsey's uncommon specificity in his biblical citations. "My god! This guy must know his  bible and therefore be an expert"!.  But what they all miss was Lindsey's preternatural penchant for finding odd historical coincidences and integrating them with other world events, along with a good dollop of numerology and plain old happenstance. Voila! Apocalytic predictions - and from the bible no less!

For example, as Lindsey tells it this "beast' could not emerge without the emergence of Israel first. One had to follow the other because Israel (specifically the valley of Megiddo) would mark the location of the final showdown between good and evil.  Since that birth occurred in 1948, and Lindsey designated one (biblical) generation = 40 years he reckoned that the Antichrist would make his first public appearance in 1988, and the Tribulation (the alleged 7-year period for his reign) would commence then. Everyone in the Antichrist’s dominion would have to be marked with the magic letters ‘666’ which Lindsey surmised would function like an electronic code similar to the product UPC codes triggered at checkout in a supermarket to get prices. Without this special numerical code, no one would be able to buy, trade or work and hence couldn’t’ survive.

This tableaux also fit in with Lindsey’s other interpretations, including for the  entity with “ten heads” mentioned in Revelation, given the European Union featured ten members at the time. Thus, the Antichrist was to arise directly out of what was the original Roman Empire, and also be in charge of rebuilding the Temple in Jerusalem (atop one of Islam’s holiest sites). The price for was to be worshipped by the Jews.

According to Lindsey’s timeline, this meant the consummation of the world would arrive within seven years of 1988, or 1995. In other words, the "Rapture" would have to take place before the Tribulation (1987 or early 1988) and Christ would march into the world for his “Second Coming” by late 1995.

Of course, none of that happened! There was no Antichrist, no Tribulation, no Rapture, and the European Union now sports 27 members. When you try to pin these knuckleheads on exactly where and when this "Antichrist" will eventually arise, their only retort is that "the time and place is known only by God". Typical, convenient and bunk. So no wonder a Trump-worshipping punk like Reed would try to now disown all the apocalyptic hubbub.

Lindsey himself has now been long forgotten except by the fundie believers and lunatics with their 5-star reviews on amazon who still think he ought to have his own bible. But hey – a good fiction idea like the Antichrist never dies – so Tim LaHaye and friends since picked it up and integrated it into their Left Behind horse pocky or what I call “The Dark Materials” series for the slow of wit and lesser imagination.

Reed insists in his WSJ piece: 

"The media routinely caricatures evangelical support for Israel as a function of their eschatology which was why two NY Times reporters found it impossible to disentangle pro-Israel attitudes from apocalyptic predictions."

But if he read Hal Lindsey's book he'd see why - because that is what millions of  Evangelicals believe! If Ralph Reed is truly convinced that evangelicals pining for Israel's role in Armageddon (and the return of Christ) is a "liberal fantasy" and overblown - he needs to check out the hundreds of rave reviews of Lindsey's book on Amazon.

See Also:

Excerpt:

While the evangelicals huff and puff and praise Israel, they only do it solely for their own selfish reasons and rationalizations: because it is to be (according to their Bibles) the stage where Armageddon plays out and J.C. Superstar returns in his illustrious “Second Coming”. Believe me, if Israel lacked eschatological import, the Fundies wouldn't give it the time of day, and indeed, would have it listed under assorted "False doctrines" pages of their blogs. The only reason they pay any attention is because of biblical references that allegedly support a 2nd Coming.

And:

And:


And: