Some Basic Elements Of Abstract Algebra - Part Three

Definition: A homomorphism φ:  R -> R' from one ring to another is a map which is compatible with the laws of composition and which carries 1 to 1, i.e. a map such that: φ(a + b) =    φ(a ) + φ (b)  and  φ(a b) =   φ(a) φ (b)

for all a,b Î  R.

An isomorphism of rings is a bijective homomorphism.  If there is an isomorphism R  -> R', the two rings are said to be isomorphic.

Def.: Isomorphism: An isomorphism of G onto H [(G, o), (H, o)] is a bijective homomorphism.

Example: H = P = {x Î R: x > 0}         (P, x)

Let G = (0, 1, 2, 3) for the operation (o) which is addition in Z₄

Let H = (2, 4, 6, 8) for the operation (o) which is multiplication in Z₁₀

Definition: Congruence modulo I.

Let S be a commutative ring. Let I be an ideal in S. We write  a  ≡  b modulo I for the two elements a, b    Î  S if

a   Î [ b ]. Hence,  b   Î [ a ].

Definition:  S is a commutative ring and I is an ideal. Then let S/ I denote  a quotient ring and the set of congruence classes modulo I. 

Perhaps the most critical aspect of ideals to note is that: “an ideal is to a ring as a normal subgroup is to a group”.  Also, in a similar manner to groups, one can have left and right ideals.

Let   a   Î  S,   a commutative ring. 

Then the set (a) defined by: (a) = {ax: ,  x Î  S }  is an ideal.

Now, let S be a commutative ring and I be an ideal in S. For each element a  Î  S let [a] be the subset of S defined by:

[a]  = I  + a = {a + j:  j   Î  S}

Further, let s be a subring of a ring S satisfying:

[a]  s  ⊆     s   and s[a]  ⊆   s  for all  a   Î  S is an ideal (or two-sided ideal of S). Also, a subring s of S satisfying [a]  s  ⊆     s    for all  a   Î  S is a left ideal of S. One satisfying  s [a] ⊆     s    for all  a   Î  S is a right ideal of S.

Suggested Problems:

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.

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

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

a)    Z   b) Z + Z   c)  Z ₅    d) Q

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

Prove that S is closed under +   and ·   

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       

See Also:

http://www.brane-space.blogspot.com/2011/02/isomorphism-between-galois-groups-and.html

And:

Brane Space: Considering A Classic Abstract Algebra Problem