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glossary of ring theory
Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.
==Definition of a ring==
;ring : A ''ring'' is a set ''R'' with two binary operations, usually called addition (+) and multiplication (×), such that ''R'' is an abelian group under addition, ''R'' is a monoid under multiplication, and multiplication is both left and right distributive over addition. Rings are assumed to have multiplicative identities unless otherwise noted. The additive identity is denoted by 0 and the multiplicative identity by 1. (''Warning'': some books, especially older books, use the term "ring" to mean what here will be called a rng; i.e., they do not require a ring to have a multiplicative identity.)
; subring : A subset ''S'' of the ring (''R'',+,×) which remains a ring when + and × are restricted to ''S'' and contains the multiplicative identity 1 of ''R'' is called a ''subring'' of ''R''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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