Bézout domain and Principal ideal ring

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Difference between Bézout domain and Principal ideal ring

Bézout domain vs. Principal ideal ring

In mathematics, a Bézout domain is a form of a Prüfer domain. In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring.

Dedekind domain

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.

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Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set.

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Integral domain

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Noetherian ring

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any chain of left (or right) ideals: there exists an n such that: Noetherian rings are named after Emmy Noether.

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Principal ideal

In the mathematical field of ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x of P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept.

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Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element.

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