# Bezout ring

An Integral domain with a unit element in which any ideal of finite type is principal. Any principal ideal ring and also any valuation ring is Bezout. A Bezout ring is integrally closed, and its localizations (i.e. its rings of fractions with respect to multiplicative systems $S$, cf. Localization in a commutative algebra) are again Bezout rings. For a finite set $a_1,\ldots,a_n$ of elements of a Bezout ring $A$ there exist a greatest common divisor (a greatest common divisor of $(a_1,\ldots,a_n)$ has the form $\sum b_i a_i$, $b_i \in A$, a so-called Bezout identity) and a least common multiple. A Noetherian ring (and even a ring that satisfies the ascending chain condition only for principal ideals) which is Bezout is a principal ideal ring. As for principal ideal rings, a module of finite type over a Bezout ring is a direct sum of a torsion module and a free module.

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#### References

[a1] | R. Gilmer, "Multiplicative ideal theory" , M. Dekker (1972) |

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Bezout ring.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Bezout_ring&oldid=34242