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Visual Group Theory, Lecture 7.1: Basic ring theory
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Visual Group Theory, Lecture 7.1: Basic ring theory
A ring is an abelian group (R,+) with a second binary operation, multiplication and the distributive law. Multiplication need not commute, nor need there be multiplicative inverses, so a ring is like a field but without these properties. Rings attempt to generalize familiar algebraic structure like the integers, reals, or complex numbers. However, many unusual things can arise: the product of nonzero elements can be zero, and ax=ay need not imply that x=y. However, these unusal properties don't happen in "integral domains", which are essentially fields without multiplicative inverses. We see a number of examples of rings, including Z, Q, R, C, Z_n, matrix rings, the Hamiltonians, and group rings.
A ring is an abelian group (R,+) with a second binary operation, multiplication and the distributive law. Multiplication need not commute, nor need there be multiplicative inverses, so a ring is like a field but without these properties. Rings attempt to generalize familiar algebraic structure like the integers, reals, or complex numbers. However, many unusual things can arise: the product of nonzero elements can be zero, and ax=ay need not imply that x=y. However, these unusal properties don't happen in "integral domains", which are essentially fields without multiplicative inverses. We see a number of examples of rings, including Z, Q, R, C, Z_n, matrix rings, the Hamiltonians, and group rings.
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