Article

# On n-simple rings

Algebra Universalis (Impact Factor: 0.44). 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.Proceedings of the American Mathematical Society 01/2004; 132(9):2543-2547. DOI:10.2307/4097370 · 0.68 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Given a ÿeld k and a positive integer n, we study the structure of the ÿnitely presented modules over the Leavitt k-algebra L of type (1; n), which is the k-algebra with a universal isomorphism i : L → L n+1 . The abelian category of ÿnitely presented left L-modules of ÿnite length is shown to be equivalent to a certain subcategory of ÿnitely presented modules over the free algebra of rank n + 1, and also to a quotient category of the category of ÿnite dimensional (over k) modules over a free algebra of rank n + 1, modulo a Serre subcategory generated by a single module. This allows us to use Schoÿeld's exact sequence for universal localization to compute the K1 group of a certain von Neumann regular algebra of fractions of L.Journal of Pure and Applied Algebra 07/2004; 191(1-2):1-21. DOI:10.1016/j.jpaa.2003.12.001 · 0.47 Impact Factor - [Show abstract] [Hide abstract]

**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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