Rings and Fields — Definitions and Examples

GRE Math Subject · Axiom Academy

LESSON Rings and Fields — Definitions and Examples Algebraic structures with two operations: addition and multiplication Additional properties a ring may or may not have: Ring with unity (unital ring): Has a multiplicative identity Division ring: Every nonzero element has a multiplicative inverse Immediate consequences of ring axioms: Commutative rings with unity (most common on GRE): -- integers mod (ring operations mod ) -- polynomials with coefficients in -- matrices over (matrix multiplication is not commutative for ) -- quaternions (a division ring that is not a field) (the prototypical integral domain) Every field is an integral domain for composite (has zero divisors) for (singular matrices are zero divisors, and not commutative) The Richest Algebraic Structure -- rationals (smallest field containing ) -- complex numbers (algebraically closed) -- finite field of elements ( prime) is a PID (principal ideal domain) when is a field The units of are exactly the nonzero constants is an integral domain but NOT a PID GRE-Style Problem: How many units does have?

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