Proper Subring Research Articles

Covering Numbers of Finite Rings

A noncyclic finite group is always equal to a union of its proper subgroups, and the minimum number of subgroups necessary to achieve this union is called the covering number of the group. Here, we investigate the analogous ideas for finite rings. We say an associative ring R with unity is coverable if it is equal to a union of its proper subrings. If this can be done using a finite number of proper subrings, then the covering number of R is the minimum number of subrings required to cover R. Not every ring is coverable, and even when R is finite it is nontrivial to decide whether R is coverable. We present a classification theorem for finite coverable semisimple rings and determine the covering number for R when R is coverable and equal to a direct product of finite fields.

Hua's fundamental theorem of geometry of rectangular matrices over EAS division rings

The fundamental theorem of geometry of rectangular matrices describes the general form of bijective maps on the space of all m×n matrices over a division ring D which preserve adjacency in both directions. This result proved by Hua in the nineteen forties has been recently improved in several directions. One can study such maps without the bijectivity assumption or one can try to get the same conclusion under the weaker assumption that adjacency is preserved in one direction only. And the last possibility is to study maps acting between matrix spaces of different sizes. The optimal result would describe maps preserving adjacency in one direction only acting between spaces of rectangular matrices of different sizes in the absence of any regularity condition (injectivity or surjectivity).A division ring is said to be EAS if it is not isomorphic to any proper subring. It has been known before that it is possible to construct adjacency preserving maps with wild behavior on matrices over division rings that are not EAS. For matrices over EAS division rings it has been recently proved that adjacency preserving maps acting between matrix spaces of different sizes satisfying a certain weak surjectivity condition are either degenerate or of the expected simple standard form. We will remove this weak surjectivity assumption, thus solving completely the long standing open problem of the optimal version of Hua's theorem.

On normal subgroups of division rings which are radical over a proper division subring

We introduce Kurosh elements in division rings based on the idea of a conjecture of Kurosh. Using this, we generalize a result of Faith in {xc[3]} and of Herstein in {xc[6]}.

Some questions concerning proper subrings

Given a ring extension \(R\subset S\) of integral domains, it is shown that if each proper subring of \(S\) containing \(R\) is integrally closed, then \(S\) is integrally closed. As an application, we show that if each proper subring of \(S\) containing \(R\) is a valuation (resp., Prufer, resp. Principal ideal) domain, then so is \(S\).

Catenarian Property of Mixed Polynomial and Power Series Rings Over a Pullback

Let T be an integral domain with a maximal ideal M, ϕ: T → K: = T/M the natural surjection, and R the pullback ϕ−1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x 1]]…[x n ]] to be catenarian, where each [x i ]] is fixed as either [x i ] or [[x i ]]. We also give a complete answer to the question of determining the field extensions k ⊂ K such that the contraction map Spec(K[x 1]]…[x n ]]) → Spec(k[x 1]]…[x n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x 1]]…[x n ]] is catenarian.

On Maximal Subrings of Commutative Rings

A proper subring S of a ring R is said to be maximal if there is no subring of R properly between S and R. If R is a noetherian domain with |R| > 2ℵ0, then | Max (R)| ≤ | RgMax (R)|, where RgMax (R) is the set of maximal subrings of R. A useful criterion for the existence of maximal subrings in any ring R is also given. It is observed that if S is a maximal subring of a ring R, then S is artinian if and only if R is artinian and integral over S. Surprisingly, it is shown that any infinite direct product of rings has always maximal subrings. Finally, maximal subrings of zero-dimensional rings are also investigated.

Sets of Elements that Pairwise Generate a Matrix Ring

In this article, we investigate the relationship between the minimum number of proper subgroups of GL(n, q) whose union is the whole GL(n, q) and the maximum number of elements that pairwise generate GL(n, q). We show that the minimum number of proper subrings of M n (q) whose union is the whole M n (q) is exactly the maximum number of elements that pairwise generate M n (q).

Rings as the Unions of Proper Subrings

We describe all possible ways how a ring can be expressed as the union of three of its proper subrings. This is an analogue for rings of a 1926 theorem of Scorza about groups.

Blocking sets of Rédei type in projective Hjelmslev planes

In this paper, we introduce Rédei type blocking sets in projective Hjelmslev planes over finite chain rings. We construct, in Hjelmslev planes over chain rings of nilpotency index 2 that contain the residue field as a proper subring, the Baer subplanes associated with this subring as Rédei type blocking sets. Two further examples of Rédei type blocking sets are given for planes over Galois rings generalizing familiar constructions in projective planes over finite fields.

The Class Group of Pullbacks

Let T be an integral domain, I a nonzero ideal of T, ϕ: T → T/I the natural ring epimorphism, D an integral domain which is a proper subring of T/I, and R = ϕ−1(D). Let k = qf(D) be the quotient field of D such that k ⊆ T/I, S = ϕ−1(k), and k* = k\\{0}. We prove that if k ⊊ T/I and if the map φ: U(T) → U(T/I)/U(D), given by u ↦ ϕ(u)U(D), is surjective, then Cl(R) = Cl(D) ⊕ Cl(S) and Pic(R) = Pic(D) ⊕ Pic(S). We also prove that the map :U(T) → U(T/I)/k*, given by u ↦ ϕ(u)k*, is surjective if and only if {x ∈ T | ϕ(x) ∈ U(T/I)} = U(T)(S\\I), if and only if the map λ:Cl(S) → Cl(T), defined by [H] ↦ [(HT) t ], is injective.

Conditions for rings to be finite or have only central zero divisors

We characterize rings having only finitely many noncentral subrings of zero divisors, and rings in which every proper noncentral subring of zero divisors is finite.

Valuation and Pseudovaluation Subrings of an Integral Domain

Let R, S be two rings. We say that R is a valuation subring of S (R is a VD in S, for short) if R is a proper subring of S and whenever x ∈ S, we have x ∈ R or x −1 ∈ R. We denote by Nu(R) the set of all nonunit elements of a ring R. We say that R is a pseudovaluation subring of S (R is a PV in S, for short) if R is a proper subring of S and x −1 a ∈ R, for each x ∈ S∖R, a ∈ Nu(R). This article deals with the study of valuation subrings and pseudovaluation subrings of a ring; interactions between the two notions are also given. Let R be a PV in S; the Krull dimension of the polynomial ring on n indetrminates over R is also computed.

On the star class group of a pullback

For the domain R arising from the construction T, M, D, we relate the star class groups of R to those of T and D. More precisely, let T be an integral domain, M a nonzero maximal ideal of T, D a proper subring of k : = T / M , φ : T → k the natural projection, and let R = φ −1 ( D ) . For each star operation * on R, we define the star operation * φ on D, i.e., the “projection” of * under φ, and the star operation ( * ) T on T, i.e., the “extension” of * to T. Then we show that, under a mild hypothesis on the group of units of T, if * is a star operation of finite type, then the sequence of canonical homomorphisms 0 → Cl * φ ( D ) → Cl * ( R ) → Cl ( * ) T ( T ) → 0 is split exact. In particular, when * = t R , we deduce that the sequence 0 → Cl t D ( D ) → Cl t R ( R ) → Cl ( t R ) T ( T ) → 0 is split exact. The relation between ( t R ) T and t T (and between Cl ( t R ) T ( T ) and Cl t T ( T ) ) is also investigated.

Generating endomorphism rings of infinite direct sums and products of modules

Let R be a ring, M a left R-module, I an infinite set, N either the direct sum or product of |I| copies of M, and E the endomorphism ring of N as a left R-module. In this note it is shown that E is not the union of a chain of |I| or fewer proper subrings, and also that given a generating set U for E as a ring, there exists a positive integer n such that every element of E is represented by a ring word of length at most n in elements of U.

A PINCHED-KRULL DOMAIN AT A PRIME IDEAL

ABSTRACT Let be an integral domain and let be a prime ideal of such that every ideal not in is -invertible. We will call such an integral domain a pinched-Krull domain at . In this paper we give some characterizations of a pinched-Krull domain which are analogs of a Krull domain. We also show that is a pinched-Krull domain at if and only if is a pinched-Krull domain at ; and that is a Krull domain if and only if is a pinched-Krull domain at . Let be a nonzero maximal ideal of an integral domain , the natural ring epimorphism, a proper subring of , and . It is proved that if and are Krull domains, then is a pinched-Krull domain at . As an application we give non-Mori domains whose prime -ideals are of finite type. At the end, we study a pinched-Krull domain at a divided prime ideal.

On the class group of a pullback

Let T be an integral domain, I a nonzero proper ideal of T, and ϕ : T→T/I the natural projection. For an integral domain D which is a proper subring of T/ I, R= ϕ −1( D) is then a subring of T. We investigate the class group Cl( R), the Picard group Pic( R), and the local class group G( R) of the pullback R.

Intersections of Matrix Algebras and Permutation Representations of PSL(n, q)

If G is a group, H a subgroup of G, and Ω a transitive G-set we ask under what conditions one can guarantee that H has a regular orbit (=of size |H|) on Ω. Here we prove that if PSL(n,q)⊆G⊆PGL(n,q) and H is cyclic then H has a regular orbit in every non-trivial G-set (with few exceptions). This result is obtained via a mixture of group theoretical and ring theoretical methods: Let R be the ring of all n×n matrices over the finite field F and let Z be the subring of scalar matrices. We show that if A and M are proper subrings of R containing Z, and if A is commutative and semisimple, then there exists an element x∈SL(n,F) such that xAx−1∩M=Z or n=2=|F|.

Involutions on Rectangular Jordan Pairs

Jordan triple systems are equivalent to Jordan pairs with involution. In recent work with D'Amour on triples of Clifford type we described involutions on pairs M1,q(Δ). Generalizing these results, in this paper we describe all involutions on nondegenerate pairs of rectangular type A(R,M,f)J having a simple artinian coordinate algebra R or, more generally, a simple unital coordinate algebra such that the form f is unital-valued: f(u,v)=1 for some u∈M+, v∈M−. The involutions are of “hermitian” type determined by an involution (anti-automorphism σ with σ2=1) on the coordinate ring, “automorphism” type determined by an automorphism σ on the coordinate ring with σ2 inner, or of “isomorphism” type determined by an isomorphism σ of the (necessarily non-artinian) coordinate ring onto a proper subring (with σ2 somewhat inner).

An Application of Jonsson Modules to Some Questions Concerning Proper Subrings.

An Application of Jonsson Modules to Some Questions Concerning Proper Subrings.

On the Cardinality of Subrings of a Commutative Ring

AbstractIf R is an uncountable commutative ring, it is shown that there exists a proper subring of R having the same cardinality as R. It is also shown that if |R| = ω is an uncountable regular cardinal, and if R1 is a subring of R containing an identity of R and such that |R1| < ω, then there exists a proper R1-subalgebra S of R such that |S| = |R|.