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subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those who define rings without requiring the existence of a multiplicative identity, a subring of ''R'' is just a subset of ''R'' that is a ring for the operations of ''R'' (this does imply it contains the additive identity of ''R''). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of ''R''). With definition requiring a multiplicative identity (which is used in this article), the only ideal of ''R'' that is a subring of ''R'' is ''R'' itself.
==Formal definition==
A subring of a ring is a subset ''S'' of ''R'' that preserves the structure of the ring, i.e. a ring with . Equivalently, it is both a subgroup of and a submonoid of .
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
■ウィキペディアで「subring」の詳細全文を読む
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