Mark each of the following as true or false. The concept of a ring homomorphism

Question

Mark each of the following as true or false. The concept of a ring homomorphism is closely connected with the idea of a factor ring. A ring homomorphism carries ideals of R into ideals of R'. A ring homomorphism is one to one if and only if the kernel is {0}. Q is an ideal of R. Every ideal in a ring is a subring of the ring. Every subring of a ring is an ideal in the ring. Every quotient ring of every commutative ring is again a commutative ring. The rings Z/4Z and Z_4 are isomorphic. An ideal I in a ring R with unity 1 is all of R if and only if 1 epsilon I. The concept of an ideal is to the concept of a ring as the concept of a normal subgroup is to the concept of the group.

Explanation / Answer

1) True, the concept of ring homomorphism with domain R gives rise to a factor ring R/N where N is the ideal of R and also its converse is true, i.e every factor ring R/N gives rise to a homomorphism mapping R into R/N.

2) False, Let : R->R' be a homomorphism.

Then [n] is an ideal of [R] where n is ideal of R although it need not be an ideal of R'.

And if n' is an ideal of [R] or R' the -1[n'] is an ideal of R.

3) True,

Let ker g {0}. then there exist a b such that f(a) = 0 = f(b) hence g is not injective.

let ker g = {0}. Assume g(a) = g(b) then g(a) - g(b) = g(b-a) = 0.

And by assumption b - a = 0. But b - a = 0 => a = b.

=> g is injective.

hence a - b ker g. So ker g = 0.

4) False, for example Z are a subring of Q (Rational numbers) but they are not an ideal as 1*4/5 = 4/5 is not an integer.

5) False, Consider Z/10Z Z/5Z x Z/2Z This ring is decomposed as product of 2 rings using Chinese remainder theorem. Factors in such a decomposition are original ring's quotient. But also corespond to an ideal of that ring but such an ideal is not a subring..

6) False, for example Z are a subring of Q (Rational numbers) but they are not an ideal as 3*4/5 = 12/5 is not an integer.

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