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On n-simple rings

Algebra Universalis (Impact Factor: 0.55). 07/2005; 53(2):301-305. DOI: 10.1007/s00012-005-1908-2

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    ABSTRACT: It is proven that every purely infinite simple ring is an exchange ring. This result is applied to determine those Leavitt algebras that are exchange rings.
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    ABSTRACT: It is proved that a nearly simple Bezout domain is an elementary divisor ring if and only if it is 2-simple. According to Kaplansky's definition (1), a ring R is an elementary divisor ring if every matrix over R is equivalent to a diagonal matrix with condition of complete divisibility of the diagonal elements. In (2) Zabavsky proved that a simple Bezout domain is an elementary divisor ring if and only if it is 2-simple. Nearly simple domains were constructed in (3-6). We prove that a nearly simple Bezout domain is an elementary divisor ring if and only if it is 2-simple. 2 Definitions Throughout R will always denote a ring (associative, but not necessarily com- mutative) with 1 6= 0. We shall write Rn for the ring of n ×n matrices with elements in R. By a unit of ring we mean an element with two-sided inverse. We'll say that matrix is unimodular if it is the unit of Rn. We denote by GLn(R) the group of units of Rn. The Jacobson radical of a ring R is denoted by J(R). An n by m matrix A = (aij) is said to be diagonal if aij = 0 for all i 6= j. We say that a matrix A admits a diagonal reduction if there exist unimodular matrices P ∈ GLn(R), Q ∈ GLm(R) such that P AQ is a diagonal matrix. We shall call two matrices A and B over a ring R equivalent (and write A ∼ B) if there exist unimodular matrices P, Q such that B = P AQ. If every matrix over R is equivalent to a diagonal matrix (dij) with the property that every dii is a total divisor of di+1,i+1 (Rdi+1,i+1R ⊆ diiR ∩ Rdii), then R is an elementary divisor ring. We recall that a ring R is said to be right (left) Hermite if every 1 by 2 (2 by 1) matrix admits a diagonal reduction, and if both, R is an Hermite ring. By a right (left) Bezout ring we mean a ring in which all finitely generated right (left) ideals are principal, and by a Bezout ring a ring which is both right and left Bezout (1). In any simple ring the property RaR = R holds for every element a ∈ R \ {0} and some depends on a. As R is a ring with identity then there exist elements
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    ABSTRACT: An example in Szigeti and van Wyk [J. Szigeti and L. van Wyk, Subrings which are closed with respect to taking the inverse, J. Algebra 318 (2007), pp. 1068–1076] suggests that Dedekind finiteness may play a crucial role in a characterization of the structural subrings M n (θ, R) of the full n × n matrix ring M n (R) over a ring R, which are closed with respect to taking inverses. It turns out that M n (θ, R) is closed with respect to taking inverses in M n (R) if all the equivalence classes with respect to θ ∩ θ−1, except possibly one, are of a size less than or equal to p (say) and M p (R) is Dedekind finite. Another purpose of this article is to show that M n (θ, R) is Dedekind finite if and only if M m (R) is Dedekind finite, where m is the maximum size of the equivalence classes (with respect to θ ∩ θ−1). This provides a positive result for the inheritance of Dedekind finiteness by a matrix ring (albeit not a full matrix ring) from a smaller (full) matrix ring.
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