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**Orders in regular rings with minimal condition for principal right ideals.**
*(English)*
Zbl 0726.16007

The purpose of this article is to give for rings without identity an infinite dimensional extension of Goldie’s theorem about orders in semisimple Artinian rings. An element \(b\in R\) is a group inverse of \(a\in R\) if \(aba=a\), \(bab=b\) and \(ab=ba\). An element \(a\in R\) is square- cancellable in R if for all x,y\(\in R\cup \{1\}\) \(a^ 2x=a^ 2y\) implies \(ax=ay\) and \(xa^ 2=ya^ 2\) implies \(xa=ya\). The ring R is a left order in Q if R is a subring of Q and:

(i) every element \(q\in Q\) can be written as \(q=a'b\) for some a,b\(\in R\), where \(a'\) is a group inverse of a in Q;

(ii) every square-cancellable element of R has a group inverse in Q.

If Q is a ring with identity and Q is von Neuman regular then this is an order in the familiar classical sense. As an infinite dimensional analogue of a semisimple Artinian ring is considered a regular ring which satisfies the minimal condition for principal right ideals (condition \(M_ R).\)

The main result is: Theorem. The following conditions for a ring R are equivalent: (1) R is an order in a regular ring Q which satisfies \(M_ R\); (2) R is semisimple and conditions (A), (B) and their left-right duals \((A')\) and \((B')\) hold: (A) for each \(a\in R\), the left ideal Ra contains no infinite direct sum of left ideals; (B) the set \(\{\ell (a)|\) \(a\in R\}\) of left annihilators satisfies the maximal condition; (3) R is semiprime, nonsingular and satisfies conditions (A) and \((A')\). Furthermore, R is prime if and only if Q is simple. As a corollary of this theorem is deduced Goldie’s characterization of orders in semisimple Artinian rings.

(i) every element \(q\in Q\) can be written as \(q=a'b\) for some a,b\(\in R\), where \(a'\) is a group inverse of a in Q;

(ii) every square-cancellable element of R has a group inverse in Q.

If Q is a ring with identity and Q is von Neuman regular then this is an order in the familiar classical sense. As an infinite dimensional analogue of a semisimple Artinian ring is considered a regular ring which satisfies the minimal condition for principal right ideals (condition \(M_ R).\)

The main result is: Theorem. The following conditions for a ring R are equivalent: (1) R is an order in a regular ring Q which satisfies \(M_ R\); (2) R is semisimple and conditions (A), (B) and their left-right duals \((A')\) and \((B')\) hold: (A) for each \(a\in R\), the left ideal Ra contains no infinite direct sum of left ideals; (B) the set \(\{\ell (a)|\) \(a\in R\}\) of left annihilators satisfies the maximal condition; (3) R is semiprime, nonsingular and satisfies conditions (A) and \((A')\). Furthermore, R is prime if and only if Q is simple. As a corollary of this theorem is deduced Goldie’s characterization of orders in semisimple Artinian rings.

Reviewer: A.I.Kashu (Kishinev)

### MSC:

16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |

16D25 | Ideals in associative algebras |

16N60 | Prime and semiprime associative rings |

### Keywords:

Goldie’s theorem; orders in semisimple Artinian rings; left order; subring; group inverse; square-cancellable element; von Neuman regular; regular ring; minimal condition for principal right ideals; direct sum of left ideals; left annihilators; semiprime
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\textit{J. Fountain} and \textit{V. Gould}, Commun. Algebra 19, No. 5, 1501--1527 (1991; Zbl 0726.16007)

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