Subject GRE-Style: Ring Homomorphism

GRE Math Subject · Axiom Academy

EXAMPLE Subject GRE-Style: Ring Homomorphism 3 worked problems on ring homomorphisms, kernels, images, and quotient rings Key Recall: A ring homomorphism satisfies and . The kernel is always an ideal, and by the First Isomorphism Theorem. Question: Define by . What is ? The kernel consists of all polynomials with . By the Factor Theorem, if and only if divides : Therefore , the principal ideal generated by . First Isomorphism Theorem check: . The map is surjective since every integer is for a constant polynomial. Question: How many elements does the ring have? (A) 5 (B) 10 (C) 25 (D) 50 (E) Infinite First mod out by 5 to get polynomials over : In , every element reduces to with , because . Count: 5 choices for , 5 choices for : Bonus insight: Check if is irreducible over . Since , it factors as , so by CRT the ring is , not a field. Which Map Is a Ring Homomorphism? Question: Which of the following is a ring homomorphism from to ? (A) : Additive? . Yes. Multiplicative? but . No. (C) : Additive? Yes. Multiplicative? but . No (unless ). (D) : The identity map. and . Yes to both! GRE Insight: Ring homomorphisms from are extremely rigid. Any ring homomorphism (with unital) is completely determined by . The only ring homomorphism is the identity.

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