Ring theory

Reduced ring

In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring R/I is reduced if and only if I is a radical ideal. Let D be the set of all zero-divisors in a reduced ring R. Then D is the union of all minimal prime ideals. Over a Noetherian ring R, we say a finitely generated module M has locally constant rank if is a locally constant (or equivalently continuous) function on Spec R. Then R is reduced if and only if every finitely generated module of locally constant rank is projective. (Wikipedia).

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From playlist Math Basics

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From playlist How to Multiply Fractions

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From playlist How to Multiply Fractions

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From playlist Add and Subtract Fractions with Like Denominators

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From playlist Algebra

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From playlist Add and Subtract Fractions with Like Denominators

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From playlist Pre-Algebra

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Related pages

Square-free integer | Prime ideal | Quotient ring | Integral domain | If and only if | Frobenius endomorphism | Minimal prime ideal | Subring | Ideal (ring theory) | Nilpotent | Perfect field | Polynomial ring | Injective function | Projective module | Characteristic (algebra) | Mathematics | Field (mathematics) | Integer | Nilradical of a ring | Algebraic geometry | Noetherian ring | Ring (mathematics) | Ring theory | Prime number | Nicolas Bourbaki | Finitely generated module | Module (mathematics) | Commutative ring