Statements about ring homomorphisms and division rings

Question

Problem

Decide whether the following statements are false or not.

1) If $A$ is a commutative ring such that every ring homomorphism different from the null morphism $\phi:A \to A'$ is injective, then $A$ is a field.

2)If $A$ is a ring such that every homomorphism different from the null morphism $\phi:A \to A'$ is injective, then $A$ is a division ring.

In 1), I've tried to show the statement is true and I couldn't but I can't think of a counterexample.

In 2), I don't understand if the morphism can be a group morphism from $(A,+) \to (A',+)$.

I would appreciate some hints prove or disprove the statements.

Answer 1

Hints:

1. If $A$ has a non-zero non-invertible element, can you think of a suitable $\phi:A\to A/I$?
2. What do you know about 2-sided ideals of $M_{2\times2}(\Bbb{R})$?