About: Irreducible ring

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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. This article follows the convention that rings have multiplicative identity, but are not necessarily commutative.

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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 irreducible 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. (en)
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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. This article follows the convention that rings have multiplicative identity, but are not necessarily commutative. (en)
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  • Irreducible ring (en)
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