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In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways. * A (meet-)irreducible ring is one in which the intersection of two nonzero ideals is always nonzero. * A directly irreducible ring is ring which cannot be written as the direct sum of two nonzero rings. * A subdirectly irreducible ring is a ring with a unique, nonzero minimum two-sided ideal. "Meet-irreducible" rings are referred to as "irreducible rings" in commutative algebra. This article adopts the term "meet-irreducible" in order to distinguish between the several types being discussed. Meet-irreducible rings play an important part in commutative algebra, and directly irreducibe and subdirectly irreducible rings play a role in the general theory of structure for rings. Subdirectly irreducible algebras have also found use in number theory. This article follows the convention that rings have multiplicative identity, but are not necessarily commutative. ==Definitions== The terms "meet-reducible", "directly reducible" and "subdirectly reducible" are used when a ring is ''not'' meet-irreducible, or ''not'' directly irreducible, or ''not'' subdirectly irreducible, respectively. The following conditions are equivalent for a commutative ring ''R'': * ''R'' is meet-irreducible; * the zero ideal in ''R'' is irreducible, i.e. the intersection of two non-zero ideals of ''A'' always is non-zero. The following conditions are equivalent for a commutative ring ''R'': * ''R'' possesses exactly one minimal prime ideal (this prime ideal may be the zero ideal); * the spectrum of ''R'' is irreducible. The following conditions are equivalent for a ring ''R'': * ''R'' is directly irreducible; * ''R'' has no central idempotents except for 0 and 1. The following conditions are equivalent for a ring ''R'': * ''R'' is subdirectly irreducible; * when ''R'' is written as a subdirect product of rings, then one of the projections of ''R'' onto a ring in the subdirect product is an isomorphism; * The intersection of all nonzero ideals of ''R'' is nonzero. 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「Irreducible ring」の詳細全文を読む スポンサード リンク
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