Algebraic Structures: maximal ideals, homomorphisms, etc

[flinted]

I need help on these 3 problems as i am currently revising for an exam and I just cant grasp these problems. I would like to show some workings but i dont know where to begin! Your help would be greatly appreciated!!

1) Show that A={(3x,y):x,y in Z} is a maximal ideal of Z+Z

2) Are the following mappings ring homomorphisms or not? If the mapping is homomorphism, is it an isomorphism?

(a) f: Z mod10 -> Z mod10 given f(x)=2x

(b) R={ |a b|: a,b,c in Z} and g:R -> Z given by
.............|0 c|
g(|a b|)=a
....|0 c|

3) If R is a commutative ring with identity, and a1,a2,...,an in R, then define (a1,a2,...,an) to be the set {r1a1,r2a2,...,rnan: ri in R}. Prove that (a1,a2,...,an is an ideal in R. It is called the ideal generated by a1,a2,...,an.

 

[daon]

I am not as far "up" as you in mathematics, however, 2)a) is an easy one.

The function f cannot be a ring homomorphism as f(xy) = 2(xy) \(\displaystyle \neq\) (2x)(2y) = f(x)f(y). However, it is a group homomorphism under addition.

2)b) Looks to be a homomorphism, but I don't believe it is an isomorphism. In order g to be an isomorphism, it must be injective. That is, for M,N in your set R, if g(M)=g(N) then M=N. Not true... Consider the matricies:

\(\displaystyle \L M = [\begin{array} 1 & 2 \\ 0 & 4 \end{array}] \,\, \,\, and \,\, \,\, N = [\begin{array} 1 & 3 \\ 0 & 5 \end{array}]\)

g, however, is surjective (onto).

I can't help much with the other questions.

Hope that helps,
-daon

 

[flinted]

thanks very much for the reply. Yea i was thinking that was the answer i just needed confirmation. I got a solution to the first question if anybody would like to see it.

 

[daon]

I would, but you'd have to explain what an ideal is . Is Z+Z the direct sum?