19 research outputs found
The row rank of a subring of a matrix ring
No full textFor
R
R
a subring of an
n
×
n
n \times n
matrix ring
M
n
(
Δ
)
{M_n}(\Delta )
over a division ring
Δ
\Delta
, we examine an invariant called the row rank of
R
R
. Roughly speaking, the row rank of
R
R
is the largest integer
k
k
such that
R
R
contains all
k
k
-rowed matrices over a left order in
Δ
\Delta
. The row rank of
R
R
is then an integer between 0 and
n
n
; and we will see that row rank
R
≥
1
R \geq 1
means that
M
n
(
Δ
)
{M_n}(\Delta )
is the maximal left quotient ring of
R
R
, while row rank
R
=
n
R = n
signifies that
M
n
(
Δ
)
{M_n}(\Delta )
is the classical left quotient ring of
R
R
. Thus row rank provides a link between maximal and classical quotient rings for rings of this type. A description of the subrings
R
R
with row rank
R
≥
k
R \geq k
is obtained which subsumes and generalizes earlier theorems of Faith-Utumi and Zelmanowitz, respectively, for the cases row rank
R
=
n
R = n
and row rank
R
≥
1
R \geq 1
.</p
Semiprime modules with maximum conditions
No full textAbstractA module RM is semiprime if for each 0 ≠ m ϵ M there exists ƒ ϵ HomR(M, R) with (mƒ)m ≠ 0. In Section 1 semiprime artinian modules are seen to be isomorphic to finite direct sums of minimal left ideals generated by idempotents. Semiprime noetherian modules have endomorphism rings which are left orders in semisimple artinian rings; and necessary and sufficient conditions for the latter situation to occur are given in Section 3. Prime modules are defined analogously and are treated simultaneously; and the above results are actually considered in the broader milieu of Morita contexts. In Sections 4 and 5 the classical density theorem for rings with faithful minimal left ideals is generalized (with a weakened definition of density) to include semiprime rings possessing faithful finite dimensional left ideals. The method of proof covers the infinite dimensional case as well. As a consequence, the classical density theorem is extended to rings with faithful completely reducible left ideals. In Section 6, the endomorphism ring of a torsionless module over a dense ring of transformations is shown to be a ring of the same type
Injective Hulls of Torsion Free Modules
No full textIn § 1, we begin with a basic theorem which describes a convenient embedding of a nonsingular left R-module into a complete direct product of copies of the left injective hull of R (Theorem 2). Several applications follow immediately. Notably, the injective hull of a finitely generated nonsingular left R-module is isomorphic to a direct sum of injective hulls of closed left ideals of R (Corollary 4). In particular, when R is left self-injective, every finitely generated nonsingular left R-module is isomorphic to a finite direct sum of injective left ideals (Corollary 6).In § 2, where it is assumed for the first time that rings have identity elements, we investigate more generally the class of left R-modules which are embeddable in direct products of copies of the left injective hull Q of R. Such modules are called torsion free, and can also be characterized by the property that no nonzero element is annihilated by a dense left ideal of R (Proposition 12).</jats:p
Commutative Endomorphism Rings
No full textThe problem of classifying the torsion-free abelian groups with commutative
endomorphism rings appears as Fuchs’ problems in [4, Problems
46 and 47]. They are far from solved, and the obstacles to a solution appear
formidable (see [4; 5]). It is, however, easy to see that the
only dualizable abelian group with a commutative endomorphism ring is the
infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class
of prime rings R which possess a dualizable module
M with a commutative endomorphism ring. A
characterization of such rings is obtained in § 6, which as would be
expected, places stringent restrictions on the ring and the module.Throughout we will write homomorphisms of modules on the side opposite to
the scalar action. Rings will not be assumed to contain identity elements
unless otherwise indicated.</jats:p
The structure of rings with faithful nonsingular modules
No full textIt is shown that the existence of a faithful nonsingular uniform module characterizes rings which have a full linear maximal quotient ring. New information about the structure of these rings is obtained and their maximal quotient rings are constructed in an explicit manner. More generally, rings whose maximal quotient rings are finite direct sums of full linear rings are characterized by the existence of a faithful nonsingular finite dimensional module.</p
The finite intersection property on annihilator right ideals
No full textIn this article the finite intersection property on annihilator right ideals will be shown to be an adequate substitute for more stringent chain conditions on such ideals. One application of the investigation will produce a new characterization of orders in semisimple artinian rings, another will generate new classes of absolutely torsion-free rings.</p
