scholarly journals On the Radical of a Hecke–Kiselman Algebra

2020 ◽  
Author(s):  
Jan Okniński ◽  
Magdalena Wiertel
Keyword(s):  
Direct Product ◽  
Oriented Graph ◽  
Rational Functions ◽  
Matrix Algebras ◽  
Oriented Cycle ◽  
Finite Module ◽  
Finite Direct Product

Abstract The Hecke-Kiselman algebra of a finite oriented graph Θ over a field K is studied. If Θ is an oriented cycle, it is shown that the algebra is semiprime and its central localization is a finite direct product of matrix algebras over the field of rational functions K(x). More generally, the radical is described in the case of PI-algebras, and it is shown that it comes from an explicitly described congruence on the underlying Hecke-Kiselman monoid. Moreover, the algebra modulo the radical is again a Hecke-Kiselman algebra and it is a finite module over its center.

scholarly journals Matrix algebras with direct product

1977 ◽  
Vol 16 (1) ◽  
pp. 19-24
Author(s):  
Michael F. O'Reilly
Keyword(s):  
Direct Product ◽  
Matrix Algebras

The lattice of congruences on direct products of cyclic semigroups and certain other semigroups

1983 ◽  
Vol 95 (3-4) ◽  
pp. 203-214 ◽  
Author(s):  
D. C. Trueman
Keyword(s):  
Direct Product ◽  
Ideal Extension ◽  
Semimodular Lattice ◽  
Direct Products ◽  
Upper Semimodular Lattice ◽  
Lattice Of Congruences ◽  
Finite Direct Product

SynopsisLet W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.

scholarly journals Quotient rings of semiprime rings with bounded index

1982 ◽  
Vol 23 (1) ◽  
pp. 53-64 ◽  
Author(s):  
John Hannah
Keyword(s):  
Direct Product ◽  
Polynomial Identity ◽  
Semiprime Ring ◽  
Quotient Ring ◽  
Matrix Rings ◽  
Special Cases ◽  
Semiprime Rings ◽  
Strongly Regular ◽  
Quotient Rings ◽  
Finite Direct Product

We say that a ring R has bounded index if there is a positive integer n such that an = 0 for each nilpotent element a of R. If n is the least such integer we say R has index n. For example, any semiprime right Goldie ring has bounded index, and so does any semiprime ring satisfying a polynomial identity [10, Theorem 10.8.2]. This paper is mainly concerned with the maximal (right) quotient ring Q of a semiprime ring R with bounded index. Several special cases of this situation have already received attention in the literature. If R satisfies a polynomial identity [1], or if every nonzero right ideal of R contains a nonzero idempotent [18] then it is known that Q is a finite direct product of matrix rings over strongly regular self-injective rings, the size of the matrices being bounded by the index of R. On the other hand if R is reduced (that is, has index 1) then Q is a direct product of a strongly regular self-injective ring and a biregular right self-injective ring of type III ([2] and [15]; the terminology is explained in [6]). We prove the following generalization of these results (see Theorems 9 and 11).

scholarly journals Warfield p-Invariants in Abelian Group Rings of Characteristic p

2013 ◽  
Vol 31 (2) ◽  
pp. 183
Author(s):  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Group Ring ◽  
General Strategy ◽  
Group Rings ◽  
Finitely Generated ◽  
Characteristic P ◽  
Commutative Group Ring ◽  
Normalized Units ◽  
Finite Direct Product

We calculate Warfield p-invariants Wα,p(V (RG)) of the group of normalized units V (RG) in a commutative group ring RG of prime char(RG) = p in each of the following cases: (1) G0/Gp is finite and R is an arbitrary direct product of indecomposable rings; (2) G0/Gp is bounded and R is a finite direct product of fields; (3) id(R) is finite (in particular, R is finitely generated). Moreover, we give a general strategy for the computation of the above Warfield p-invariants under some restrictions on R and G. We also point out an essential incorrectness in a recent paper due to Mollov and Nachev in Commun. Algebra (2011).

ON THE EXISTENCE OF MAXIMAL SUBRINGS IN COMMUTATIVE ARTINIAN RINGS

2010 ◽  
Vol 09 (05) ◽  
pp. 771-778 ◽  
Author(s):  
A. AZARANG ◽  
O. A. S. KARAMZADEH
Keyword(s):  
Direct Product ◽  
Characteristic Zero ◽  
Artinian Ring ◽  
Artinian Rings ◽  
Finite Direct Product

We determine entirely which Artinian rings have maximal subring. In particular, we show that an Artinian ring without maximal subring is integral over some finite subring and in particular that every Artinian ring which is uncountable or of characteristic zero has a maximal subring. We also determine when a finite direct product of rings has a maximal subring. Finally, we show that if a ring R has an Artinian maximal subring then R itself is Artinian.

scholarly journals ON THE REGULAR DIGRAPH OF IDEALS OF COMMUTATIVE RINGS

2012 ◽  
Vol 88 (2) ◽  
pp. 177-189 ◽  
Author(s):  
M. AFKHAMI ◽  
M. KARIMI ◽  
K. KHASHYARMANESH
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Direct Product ◽  
Commutative Rings ◽  
Nonzero Divisor ◽  
Vertex Set ◽  
Regular Digraph ◽  
Finite Direct Product

AbstractLet$R$be a commutative ring. The regular digraph of ideals of$R$, denoted by$\Gamma (R)$, is a digraph whose vertex set is the set of all nontrivial ideals of$R$and, for every two distinct vertices$I$and$J$, there is an arc from$I$to$J$whenever$I$contains a nonzero divisor on$J$. In this paper, we study the connectedness of$\Gamma (R)$. We also completely characterise the diameter of this graph and determine the number of edges in$\Gamma (R)$, whenever$R$is a finite direct product of fields. Among other things, we prove that$R$has a finite number of ideals if and only if$\mathrm {N}_{\Gamma (R)}(I)$is finite, for all vertices$I$in$\Gamma (R)$, where$\mathrm {N}_{\Gamma (R)}(I)$is the set of all adjacent vertices to$I$in$\Gamma (R)$.

scholarly journals Representations of the direct product of matrix algebras

2001 ◽  
Vol 169 (2) ◽  
pp. 145-160
Author(s):  
Daniele Guido ◽  
Lars Tuset
Keyword(s):  
Direct Product ◽  
Matrix Algebras

scholarly journals Regular metabelian groups of prime-power order

1970 ◽  
Vol 3 (1) ◽  
pp. 49-54 ◽  
Author(s):  
R. J. Faudree
Keyword(s):  
Direct Product ◽  
Prime Power ◽  
Prime Power Order ◽  
Metabelian Groups ◽  
Finite Direct Product

Let H be a finite metabelian p-group which is nilpotent of class c. In this paper we will prove that for any prime p > 2 there exists a finite metacyclic p-group G which is nilpotent of class c such that H is isomorphic to a section of a finite direct product of G.

scholarly journals A new characterization of Dedekind domains

1986 ◽  
Vol 28 (2) ◽  
pp. 237-239 ◽  
Author(s):  
D. D. Anderson ◽  
E. W. Johnson
Keyword(s):  
Direct Product ◽  
Principal Ideal ◽  
Prime Ideals ◽  
Integral Domains ◽  
Dedekind Domains ◽  
Finite Direct Product

Throughout this paper all rings are assumed commutative with identity. Among integral domains, Dedekind domains are characterized by the property that every ideal is a product of prime ideals. For a history and proof of this result the reader is referred to Cohen [2, pp. 31–32]. More generally, Mori [5] has shown that a ring has the property that every ideal is a product of prime ideals if and only if it is a finite direct product of Dedekind domains and special principal ideal rings (SPIRS). Rings with this property are called general Z.P.I.-rings.

On Strongly Ordered Congruences and Decompositions of Ordered Semigroups

2008 ◽  
Vol 15 (04) ◽  
pp. 589-598 ◽  
Author(s):  
Xiang-yun Xie
Keyword(s):  
Direct Product ◽  
Subdirect Product ◽  
Ordered Semigroup ◽  
Ordered Semigroups ◽  
Strongly Regular ◽  
Finite Direct Product ◽  
Regular Congruence

In this paper, we introduce the concept of a strongly ordered congruence on a directed ordered semigroup S. We prove that any strongly ordered congruence on S is a strongly regular congruence. We characterize the finite direct product, subdirect product and full subdirect product of ordered semigroups by using the concepts of strongly ordered congruence and regular congruence on an ordered semigroup S.

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