Related papers: The zero short Covering Problem for finite rings
We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM…
For any ring we propose the construction of a cover which increases the finitistic dimension on one side and decreases the finitistic dimension to zero on the opposite side. This complements recent work of Cummings.
We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k)…
For D an infinite set, k>1 and W the set of k-sets from D, there is a natural closed permutation group G_k which is a non-split extension of \mathbb{Z}_2^W by \Sym(D). We classify the closed subgroups of G_k which project onto \Sym(D)$. The…
Completely regular codes with covering radius $\rho=1$ must have minimum distance $d\leq 3$. For $d=3$, such codes are perfect and their parameters are well known. In this paper, the cases $d=1$ and $d=2$ are studied and completely…
A covering of a group is a finite set of proper subgroups whose union is the whole group. A covering is minimal if there is no covering of smaller cardinality, and it is nilpotent if all its members are nilpotent subgroups. We complete a…
If $R$ is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) $R$ is unit-regular, (2) every factor ring of $R$ is directly finite, (3) the abelian group $K_0(R)$ is free and admits a basis which…
We present new self-gravitating solutions in five dimensions that describe circular strings, i.e., rings, electrically coupled to a two-form potential (as e.g., fundamental strings do), or to a dual magnetic one-form. The rings are…
Our main result states that a finite semiring of order >2 with zero which is not a ring is congruence-simple if and only if it is isomorphic to a `dense' subsemiring of the endomorphism semiring of a finite idempotent commutative monoid. We…
In this paper we obtain sharp results for Waring's problem over general finite rings, by using a combination of Artin-Wedderburn theory and Hensel's lemma and building on new proofs of analogous results over finite fields that are achieved…
We characterize which complete local (Noetherian) rings T containing the rationals are the completion of a countable excellent local ring S. We also discuss the possibilities for the map from the minimal prime ideals of T to the minimal…
Let $R$ be a regular ring containing a field $k$. Let $\mathbf{x} = x_1, \ldots, x_r$ be a regular sequence in $R$ such that $R/(\mathbf{x})$ is a regular ring. Fix $m \geq 1$. Set $A_m = R/(\mathbf{x})^m$. We show that for any ideal $Q$ of…
In this paper, we construct a local ring $A$ such that the kernel of the map $G_0(A)\subq \to G_0(\hat{A})\subq$ is not zero, where $\hat{A}$ is the comletion of $A$ with respect to the maximal ideal, and $G_0()\subq$ is the Grothendieck…
We give an elementary proof of a result which is not as well known as it should be: a ring with a specified finite number of zero divisors is finite, with a precise bound on its order.
Using a result of M. Hochster and C. Huneke on $F$-rational rings a criterion for complete intersection rings of characteristic $p>0$ is presented. As an application, we give a completely different proof for an algebraic result of G.…
We construct explicit $\delta$-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these $\delta$-bracketing covers are bounded from…
In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding…
In this paper we investigate codes over finite commutative rings R, whose generator matrices are built from \$\alpha\$-circulant matrices. For a non-trivial ideal I<R we give a method to lift such codes over R/I to codes over R, such that…
We apply recent results on the rank of elements of rings to study the structure of generalized corner rings $aRa$, where $R$ is a unital ring and $a$ an element of $R$. We give a complete description of the structure of $aRa$ when $a^2$ has…
This paper studies "pro-excision" for the K-theory of one-dimensional (usually semi-local) rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite…