This volume contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held from August 10-19, 2016, at Syracuse University, Syracuse, NY. ...Included are three survey articles based on short courses in the areas of commutative algebraic groups, modular group representation theory, and thick tensor ideals of bounded derived categories. Other articles represent contributions to areas in and related to representation theory, such as noncommutative resolutions, twisted commutative algebras, and upper cluster algebras.
MacWilliams proved that every finite field has the extension property for Hamming weight which was later extended in a seminal work by Wood who characterized finite Frobenius rings as precisely those ...rings which satisfy the MacWilliams extension property. In this paper, the question of when is a MacWilliams ring quasi-Frobenius is addressed. It is proved that a right or left noetherian left 1-MacWilliams ring is quasi-Frobenius thus answering the different questions asked in 13,22. We also prove that a right perfect, left automorphism-invariant ring is left self-injective. In particular, this yields that if R is a right (or left) artinian, left automorphism-invariant ring, then R is quasi-Frobenius, thus answering a question asked in 13.
Let R be a commutative Artinian ring and let
be the compressed zero-divisor graph associated to R. The question of when the clique number
was raised by J. Coykendall, S. Sather-Wagstaff, L. ...Sheppardson, and S. Spiroff. They proved that if
(where
is the largest length of any of its chains of ideals), then
When
they gave an example of a local ring R where
is possible by using the trivial extension of an Artinian local ring by its dualizing module. The question of what happens when
was stated as an open question. We show that if
then
We first reduce the problem to the case of a local ring
We then enumerate all possible Hilbert functions of R and show that the k-vector space
admits a symmetric bilinear form in some cases of the Hilbert function. This allows us to relate the orthogonality in the bilinear space
with the structure of zero-divisors in R. For instance, in the case when
is principal and
we show that R is Gorenstein if and only if the symmetric bilinear form on
is non-degenerate. Moreover, in the case when
our techniques also yield a simpler and shorter proof of the finiteness of
avoiding, for instance, the Cohen structure theorem.
On simple-direct modules Büyükaşık, Engin; Demir, Özlem; Diril, Müge
Communications in algebra,
20/2/1/, Letnik:
49, Številka:
2
Journal Article
Recenzirano
Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and ..."simple-direct-projective," respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right H-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.
A restricted artinian ring is a commutative ring with an identity in which every proper homomorphic image is artinian. Cohen proved that a commutative ring
is restricted artinian if and only if it is ...noetherian and every nonzero prime ideal of
is maximal. Facchini and Nazemian called a commutative ring isoartinian if every descending chain of ideals becomes stationary up to isomorphism. We show that every proper homomorphic image of a commutative noetherian ring
is isoartinian if and only if
has one of the following forms:
(a)
is a noetherian domain of Krull dimension one which is not a principal ideal domain;
(b)
, where each
is a principal ideal domain and each
is an artinian local ring (either
or
may be zero);
(c)
is a noetherian ring of Krull dimension one, simple unique minimal prime ideal
, and
is a principal ideal domain. As an application of our result, we describe commutative rings whose proper homomorphic images are principal ideal rings. Some relevant examples are provided.
Let
R
be a ring. An
R
-module
M
is called biretractable if Hom
R
(
M
/
K
,
N
)≠ 0 for any proper submodule
K
of
M
and any nonzero submodule
N
of
M
. It is shown that being biretractable is preserved ...by taking direct summands but it is not preserved under submodules, factor modules, direct sums and extensions. A characterization of biretractable modules
M
with Rad(
M
)≠
M
or Soc(
M
)≠ 0 is provided. The structure of biretractable modules over Dedekind domains is fully determined. It is proved that the right
R
-module
R
R
is biretractable if and only if
R
has a unique simple right
R
-module (up to isomorphism) and has an essential right socle. Also, the class of rings
R
for which every
R
-module is biretractable, is shown to be precisely that of right semi-artinian right max rings having a unique simple right
R
-module (up to isomorphism).
"Let A be a class of right R-modules that is closed under isomorphisms, and let M be a right R-module. Then M is called A -C3 if, whenever N and K are direct summands of M with N ∩K = 0 and K ∈ A , ...then N ⊕K is also a direct summand of M; M is called an A -C4 module, if whenever M = A⊕B where A and B are submodules of M and A ∈ A , then every monomorphism f : A → B splits. Some characterizations and properties of these classes of modules are investigated. As applications, some new characterizations of semisimple artinian rings, right V-rings, quasi-Frobenius rings and von Neumann regular rings are given."
In this paper we work with partial actions and unital twisted partial actions of groups. We investigate ring theoretic properties of partial crossed products and partial skew group rings as ...artinianity, noetherianity, perfect property, semilocal property, semiprimary property and Krull dimension. Moreover, we consider triangular matrix representation of partial skew group rings, weak and global dimensions of partial crossed products. Finally we study when the partial crossed products are Frobenius algebras.
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