Ideals and Quotient Rings

GRE Math Subject · Axiom Academy

LESSON Ideals and Quotient Rings The ring-theoretic analog of normal subgroups and quotient groups Just as normal subgroups let us form quotient groups, ideals let us form quotient rings. The absorption property is the key difference from a subring. An ideal "absorbs" multiplication by any ring element -- it swallows products. Trivial ideals: Every ring has at least two ideals: In fact, every ideal of is principal . This makes a PID (principal ideal domain). Maximal ideals: for prime only Addition and multiplication are done mod Application: is a field is maximal in is prime. The Bridge Between Homomorphisms and Quotients Example: The evaluation map defined by : This is a surjective ring homomorphism Problem 1: Is an integral domain? Problem 2: How many ring homomorphisms exist from to ?

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