← MATH 01340 · Modern Algebra I
MATH 01340 · Rings & fields (intro)
Ring axioms
Definition
A ring has two operations (+,·): (R,+) abelian group, · associative and distributive over +. Many texts require 1 (unity). Commutative rings have commutative multiplication.
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Examples: ℤ, ℤₙ, matrices Mₙ(ℝ) (noncommutative), polynomials F[x].
0 annihilates: 0·a=0. Units are multiplicative inverses when they exist.
Subrings inherit structure; ideals play the role analogous to normal subgroups for quotients.
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1.Verify ℤ₆ is a ring; find its units.