Question

1.
A homomorphism from a ring R into a ring R' is said to be isomorphism if
(a) is one to one
(b) o is onto
(c) both (a) and (b)
(d) none​

Answer 1

Answer:

l explain uh so I think this much is good ❤️

Step-by-step explanation:

Endomorphisms, isomorphisms, and automorphisms

One can prove that a ring homomorphism is an isomorphism if and only if it is bijective as a function on the underlying sets. If there exists a ring isomorphism between two rings R and S, then R and S are called isomorphic.

Answer 2

Homomorphism and isomorphism:

Explanation:

• Homomorphism:
• In algebra, a homomorphism may be a structure-preserving map between two algebraic structures of the identical type (such as two groups, two rings, or two vector spaces).
• Ring homomorphism:
• A ring homomorphism (or a hoop map for short) could be a function f is from  A to B such that: (a) For all x, y ∈ A, f(x + y) = f(x) + f(y). (b) For all x, y ∈ A,then the operation should be preserved which means  f(xy) = f(x)f(y). Usually, we require that if A and B are rings with 1, then (c) f(A)=B.

• Ring Isomorphism:
• Isomorphic rings have all their ring-theoretic properties identical. One such ring will be thought to be "the same" because the other. The inverse map of the bijection f is additionally a hoop homomorphism.
• A homomorphism from a hoop R into a hoop R' is alleged to be isomorphism :
• One can prove that a hoop homomorphism is an isomorphism if and as long as it's bijective as a function on the underlying sets. If there exists a hoop isomorphism between two rings A and B, then A and B are called isomorphic.

Hence, C is the correct option.