Let be a ring with 1 and let and be (left, unital) -modules. Prove that is injective if and only if and are injective.
Recall Baer’s Criterion: an -module is injective if and only if for every left ideal and every -module homomorphism , there exists an -module homomorphism such that .
Suppose and are injective. Let be a left ideal and let be an -module homomorphism. Letting and denote the first and second coordinate projections, and are -module homomorphisms. By Baer’s Criterion, there exist -module homomorphisms and such that and . Define by . Certainly is an -module homomorphism. Moreover, . Thus is injective.
Suppose is injective. Let be a left ideal and let and be -module homomorphisms. Define by . Certainly is an -module homomorphism. By Baer’s Criterion, there exists an -module homomorphism extending . Now let and ; certainly and are -module homomorphisms, and if then . Thus and are injective.