Computing the symmetric ring of quotients.

*(English)*Zbl 0605.16003Let R be an associative prime ring and let \(Q_{\ell}(R)\) be the Martindale ring of quotients [W. S. Martindale, J. Algebra 12, 576- 584 (1969; Zbl 0175.031)]. In this paper the author briefly discusses the Martindale ring of quotients, introduces the symmetric Martindale ring of quotients \(Q_ s(R)\) and obtains some basic properties. However, for the most part he computes the symmetric ring of quotients for free algebras and large classes of group algebras. These two problems are posed by S. Montgomery.

The prime ring R is symmetrically closed if \(Q_ s(R)=R\). For example, if K is a field and \(R=K<x,y,...>\) is a free K-algebra on at least two generators, then R is symmetrically closed (Theorem 2.5). Or, if every nontrivial normal subgroup of the group G is uncountable, then the group ring K[G] over the field K is symmetrically closed (Theorem 5.6). Let F be a field extension of K. Then K[G] is symmetrically closed if and only if F[G] is symmetrically closed (Theorem 8.1). Furthermore \(Q_{\ell}(K[G])=K[G]\) if and only if \(Q_{\ell}(F[G])=F[G]\) (Theorem 8.2).

In fact, most of this paper is devoted to the study of group algebras. Furthermore for the sake of simplicity the author does not always offer the best possible result. But he notes: ”It is clear that much of what we do here extends to crossed products and to more general groups. However, these extensions would necessarily complicate our arguments and are therefore more appropriately put off to a later paper.”

The prime ring R is symmetrically closed if \(Q_ s(R)=R\). For example, if K is a field and \(R=K<x,y,...>\) is a free K-algebra on at least two generators, then R is symmetrically closed (Theorem 2.5). Or, if every nontrivial normal subgroup of the group G is uncountable, then the group ring K[G] over the field K is symmetrically closed (Theorem 5.6). Let F be a field extension of K. Then K[G] is symmetrically closed if and only if F[G] is symmetrically closed (Theorem 8.1). Furthermore \(Q_{\ell}(K[G])=K[G]\) if and only if \(Q_{\ell}(F[G])=F[G]\) (Theorem 8.2).

In fact, most of this paper is devoted to the study of group algebras. Furthermore for the sake of simplicity the author does not always offer the best possible result. But he notes: ”It is clear that much of what we do here extends to crossed products and to more general groups. However, these extensions would necessarily complicate our arguments and are therefore more appropriately put off to a later paper.”

Reviewer: S.V.Mihovski

##### MSC:

16P50 | Localization and associative Noetherian rings |

16S34 | Group rings |

16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |

16N60 | Prime and semiprime associative rings |

##### Keywords:

Martindale ring of quotients; symmetric Martindale ring of quotients; free algebras; group algebras; prime ring; crossed products
Full Text:
DOI

##### References:

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