Article

# On n-simple rings

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

- [Show abstract] [Hide abstract]

**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. · 0.63 Impact Factor - [Show abstract] [Hide abstract]

**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.Linear and Multilinear Algebra 02/2011; 59(2):221-227. · 0.70 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 elementsBuletinul Academiei de Ştiinţe a Republicii Moldova. Matematica. 01/2006; 2006(3).

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.