In this work, we present a number of generator matrices of the form I2n|τ2(v), where I2n is the 2n×2n identity matrix, v is an element in the group matrix ring M2(R)G and where R is a finite ...commutative Frobenius ring and G is a finite group of order 18. We employ these generator matrices and search for binary 72,36,12 self-dual codes directly over the finite field F2. As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings.
Let
be a unital associative ring. Our motivation is to prove that left derivations in column finite matrix rings over
are equal to zero and demonstrate that a left derivation
in the infinite upper ...triangular matrix ring
is determined by left derivations
in
satisfying
for any
, where
The similar results about Jordan left derivations are also obtained when
is 2-torsion free.
The classical Morita Theorem for rings established the equivalence of three statements involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule ...homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings.
Real Nullstellensatz is a classical result from Real Algebraic Geometry. It has recently been extended to quaternionic polynomials by Alon and Paran 1. The aim of this paper is to extend their ...Quaternionic Nullstellensatz to matrix polynomials. We also obtain an improvement of the Real Nullstellensatz for matrix polynomials from 4 in the sense that we simplify the definition of a real left ideal. We use the methods from the proof of the matrix version of Hilbert's Nullstellensatz 5 and we obtain their extensions to a mildly non-commutative case and to the real case.
Abstract The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either $\lceil \log _2 n\rceil $ or $\lceil ...\log _2 n\rceil +1$ , where n is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.
Study of Morita contexts Tang, Gaohua; Li, Chunna; Zhou, Yiqiang
Communications in algebra,
04/2014, Letnik:
42, Številka:
4
Journal Article
Recenzirano
This article concerns mainly on various ring properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a generalized ...matrix ring over a ring to satisfy a certain ring property which is among being semilocal, semiperfect, left perfect, semiprimary, semipotent, potent, clean, strongly π-regular, semiregular, etc. Many known results on a formal triangular matrix ring are extended to a Morita context or a trivial Morita context. Some questions on this subject raised by Varadarajan in
22
are answered.
Suppose that F is a finite field and R=Mn(F) is the ring of n-square matrices over F. Here we characterize when the Cayley graph of the additive group of R with respect to the set of invertible ...elements of R, called the unitary Cayley graph of R, is well-covered. Then we apply this to characterize all finite rings with identity whose unitary Cayley graph is well-covered or Cohen-Macaulay.
We obtain algorithmically effective versions of the dense lattice sphere packings constructed from orders in Q-division rings by the first author. The lattices in question are lifts of suitable codes ...from prime characteristic to orders O in Q-division rings and we prove a Minkowski-Hlawka type result for such lifts. Exploiting the additional symmetries under finite subgroups of units in O , we show that this leads to effective constructions of lattices approaching the best known lower bounds on the packing density Δ n in a variety of new dimensions n . This unifies and extends a number of previous constructions.
We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not a ring R being ...nil-clean implies that the matrix ring
M
n
(
R
)
over R is also nil-clean for all
n
≥
1
is paralleling to the corresponding implication for (abelian, local) periodic rings. Besides, we study when the endomorphism ring
E
(
G
)
of an abelian group G is periodic. Concretely, we establish that
E
(
G
)
is periodic exactly when G is finite as well as we find a complete necessary and sufficient condition when the endomorphism ring over an abelian group is strongly m-nil clean for some natural number m thus refining an "old" result concerning strongly nil-clean endomorphism rings. Responding to a question when a group ring is periodic, we show that if R is a right (resp., left) perfect periodic ring and G is a locally finite group, then the group ring RG is periodic, too. We finally find some criteria under certain conditions when the tensor product of two periodic algebras over a commutative ring is again periodic. In addition, some other sorts of rings very close to periodic rings, namely the so-called weakly periodic rings, are also investigated.
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