Proper Subring Research Articles

The covering numbers of rings

A cover of an associative (not necessarily commutative nor unital) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number σ(R) of R is the cardinality of a minimal cover, and a ring R is called σ-elementary if σ(R)<σ(R/I) for every nonzero two-sided ideal I of R. If R is a ring with unity, then we define the unital covering number σu(R) to be the size of a minimal cover of R by subrings that contain 1R (if such a cover exists), and R is σu-elementary if σu(R)<σu(R/I) for every nonzero two-sided ideal of R. In this paper, we classify all σ-elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all σu-elementary rings and prove σu(R)=σ(R) for every σu-elementary ring R. We also prove that, if R is a ring without unity with a finite cover, then there exists a unital ring R′ such that σ(R)=σu(R′), which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, ifE(N):={m:m⩽N,σ(R)=m for some ring R}, then we show that |E(N)|=Θ(N/log⁡N), which proves that almost all integers are not covering numbers of a ring.

A new infinite family of σ-elementary rings

A cover of an associative (not necessarily commutative nor unital) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number of R is the cardinality of a minimal cover, and a ring R is called σ-elementary if for every nonzero two-sided ideal I of R. In this paper, we provide the first examples of σ-elementary rings R that have nontrivial Jacobson radical J with R/J noncommutative, and we determine the covering numbers of these rings.

Trace-based cryptanalysis of cyclotomic $R_{q,0}\times R_q$-PLWE for the non-split case

We describe a decisional attack against a version of the PLWE problem in which the samples are taken from a certain proper subring of large dimension of the cyclotomic ring $\mathbb{F}_q[x]/(\Phi_{p^k}(x))$ with $k&gt;1$ in the case where $q\equiv 1\pmod{p}$ but $\Phi_{p^k}(x)$ is not totally split over $\mathbb{F}_q$. Our attack uses the fact that the roots of $\Phi_{p^k}(x)$ over suitable extensions of $\mathbb{F}_q$ have zero-trace and has overwhelming success probability as a function of the number of input samples. An implementation in Maple and some examples of our attack are also provided.

Finite unitary rings all of whose groups of units of all their subrings except of the ring itself are solvable

Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper, we classify these rings and show that all finite rings of order [Formula: see text] for [Formula: see text] and some of order [Formula: see text] are in this class of rings.

Countably coverable rings

Let R be an associative ring. Then R is said to be coverable provided R is the union of its proper subrings (which we do not require to be unital even if R is so). One verifies easily that R is coverable if and only if R is not generated as a ring by a single element. In case R can be expressed as the union of a finite number of proper subrings, the least such number is called the covering number of R. Covering numbers of rings have been studied in a series of recent papers. The purpose of this note is to study rings which can be covered by a countable collection of subrings.

Certain towers of ramified minimal ring extensions of commutative rings, II

Let [Formula: see text] be an odd prime number and [Formula: see text]. Assume that either [Formula: see text]i[Formula: see text] [Formula: see text] or [Formula: see text]ii[Formula: see text] [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text] [Formula: see text]respectively, assume that [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text]. Then there exist exactly five [Formula: see text]respectively, exactly seven[Formula: see text] isomorphism classes of rings [Formula: see text] for which there exists a tower [Formula: see text] of ramified [Formula: see text]integral minimal[Formula: see text] ring extensions such that [Formula: see text] is the only ring properly contained between [Formula: see text] and [Formula: see text]. We produce a set of isomorphism class representatives of such [Formula: see text] and, for each such [Formula: see text], we show that, up to isomorphism, the corresponding ring [Formula: see text] is the idealization [Formula: see text]. One consequence, for each integer [Formula: see text] whose prime-power factorization [Formula: see text] (with pairwise distinct prime numbers [Formula: see text] and positive integers [Formula: see text]) satisfies [Formula: see text] for all [Formula: see text], is a classification, up to isomorphism, of the rings that have cardinality [Formula: see text] and exactly two proper (unital) subrings.

Maximal non-Prüfer and maximal non--Prüfer rings

Let R be a commutative ring with unity. Let denotes the set of all rings R such that is a divided prime ideal. The notion of maximal non-Prüfer ring and maximal non--Prüfer ring is introduced which generalize the concept of maximal non-Prüfer subrings of a field. A proper subring R of a ring S is said to be a maximal non-Prüfer subring of S if R is not a Prüfer ring but every subring of S which contains R properly is a Prüfer ring. A proper subring R of a ring S is said to be maximal non--Prüfer subring of S if R is not a -Prüfer ring but every subring of S which contains R properly is a -Prüfer ring. We study the properties of maximal non-Prüfer subrings and maximal non--Prüfer subrings of a ring in class Characterizations of a ring in class to be a maximal non-Prüfer ring and maximal non--Prüfer ring are given. Examples of a maximal non--Prüfer subring which is not a maximal non-Prüfer subring and a maximal non-Prüfer subring which is not a maximal non--Prüfer subring are also given to strengthen the concept.

Isonoetherian power series rings II

We study the isonoetherian and “ACC d on ideals” properties in power series rings, polynomial rings, rings of the form (where I is an ideal of a commutative unitary ring R), (where D is an integral domain with quotient field K), Nagata’s idealization ring and almost pseudo valuation domain. We point out three mistakes in two works. Let R be a quasi-local proper subring of an integral domain S such that R is integrally closed in S. We prove that is u-isonoetherian if and only if R satisfies ACC d on ideals, S is Noetherian and there exists a maximal ideal M of S that is a divided prime ideal of R and R/M is a rank one discrete valuation domain with quotient field S/M. We prove that the ring is isonoetherian if and only if R is a rank one discrete valuation domain with quotient field S. We deduce an example of a u-isonoetherian ring extension such that the ring is not isonoetherian.

On the subring generated by commutators

Let [Formula: see text] be a ring. By the notation [Formula: see text] we denote the additive subgroup of [Formula: see text] generated by all [Formula: see text] in [Formula: see text]. In this work, we partially generalize a result due to Herstein [I. N. Herstein, Topics in Ring Theory (University of Chicago Press, 1969)] showing that if [Formula: see text], then the subring generated by [Formula: see text] is equal to [Formula: see text]. This result implies that [Formula: see text] cannot be a proper subring of [Formula: see text].

Covering numbers of commutative rings

A cover of a unital, associative (not necessarily commutative) ring R is a collection of proper subrings of R whose set-theoretic union equals R . If such a cover exists, then the covering number σ ( R ) of R is the cardinality of a minimal cover , and a ring R is called σ -elementary if σ ( R ) < σ ( R / I ) for every nonzero two-sided ideal I of R . In this paper, we show that if R has a finite covering number, then the calculation of σ ( R ) can be reduced to the case where R is a finite ring of characteristic p and the Jacobson radical J of R has nilpotency 2. Our main result is that if R has a finite covering number and R / J is commutative (even if R itself is not), then either σ ( R ) = σ ( R / J ) , or σ ( R ) = p d + 1 for some d ⩾ 1 . As a byproduct, we classify all commutative σ -elementary rings with a finite covering number and characterize the integers that occur as the covering number of a commutative ring .

A note on $\lambda$-domains and $\Delta$-domains

Let $R$ be an integral domain. Then $R$ is said to be a $\lambda$-domain if the set of all overrings of $R$ is linearly ordered by inclusion. If $R_1 + R_2$ is an overring of $R$ for each pair of overrings $R_1, R_2$ of $R$, then $R$ is said to be a $\Delta$-domain. We show that if $R\subset T$ is an extension of integral domains such that each proper subring of $T$ containing $R$ is a $\lambda$-domain (resp., $\Delta$-domain), then $T$ is a $\lambda$-domain (resp., $\Delta$-domain under some conditions). Moreover, the pair $(R, T)$ is a residually algebraic pair. Two new ring theoretic properties, namely $\lambda$-property of domains and $\Delta$-property of domains are introduced and studied.

Finite unitary rings in which every subring is commutative, are commutative

We characterize the structure of finite unitary rings [Formula: see text] in which every proper subring is commutative and show that [Formula: see text] is a commutative ring.

English

Let R be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring R of an integral domain S is called a maximal non valuation domain in S if R is not a valuation subring of S, and for any ring T such that R ⊂ T ⊂ S, T is a valuation subring of S. For a local domain S, the equivalence of an integrally closed maximal non VD in S and a maximal non local subring of S is established. The relation between dim(R, S) and the number of rings between R and S is given when R is a maximal non VD in S and dim(R, S) is finite. For a maximal non VD R in S such that R ⊂ R'S ⊂ S and dim(R, S) is finite, the equality of dim(R, S) and dim(R'S, S) is established.

Regular sequences in the subrings of C(X)

We show that the intermediate subalgebras between $C^*(X)$ and C(X) do not contain regular sequences with length $\geq$ 2. This shows that depth(A(X)) $\leq$ 1 for each intermediate subalgebra A(X) between $C^*(X)$ and C(X). Whenever an intermediate subalgebra A(X) is proper, i.e. A(X) $\neq$ C(X), we observe that the depth of A(X) is exactly 1. Using this, it turns out that depth($C^*(X)$) = 0 if and only if $X$ is a pseuodocompact almost $P$ -space. The regular sequences in the subrings of the form $I + \mathbb{R}$ of C(X), where $I$ is a $z$-ideal of C(X), are also investigated and we have shown that the length of regular sequences in such rings is at most 1. In contrast to the depth of intermediate subalgebras, we see that the depth of a proper subring of the form $I + \mathbb{R}$ may be zero. Finally, regular sequences of extension rings of C(X) are also studied and some examples of subrings of C(X) are given with depths different from the depth of C(X).

Covering numbers of upper triangular matrix rings over finite fields

A cover of a finite ring R is a collection of proper subrings {S1,…,Sm} of R such that R= ⋃i=1mSi. If such a collection exists, then R is called coverable, and the covering number of R is the cardinality of the smallest possible cover. We investigate covering numbers for rings of upper triangular matrices with entries from a finite field. Let Fq be the field with q elements and let Tn(Fq) be the ring of n×n upper triangular matrices with entries from Fq. We prove that if q≠4, then T2(Fq) has covering number q+1, that T2(F4) has covering number 4, and that when p is prime, Tn(Fp) has covering number p+1 for all n≥2.

Maximal Non $$\phi $$-Chained Rings and Maximal Non Chained Rings

Let R be a commutative ring with unity. The notion of maximal non chained subrings of a ring and maximal non $$\phi $$ -chained subrings of a ring is introduced which generalizes the concept of maximal non valuation subrings of a domain. A ring R is said to be a maximal non chained (resp., $$\phi $$ -chained) subring of S if R is a proper subring of S, R is not a chained (resp., $$\phi $$ -chained) ring and every subring of S which contains R properly is a chained (resp., $$\phi $$ -chained) ring. We study the properties and characterizations of a maximal non chained ( $$\phi $$ -chained) subring of a ring. Examples of a maximal non $$\phi $$ -chained subring which is not a maximal non chained subring and a maximal non chained subring which is not a maximal non $$\phi $$ -chained subring are also given to strengthen the concept.

Valuative rings

For a pair of commutative rings [Formula: see text] with the same identity, [Formula: see text] is said to be [Formula: see text]-valuative if for each nonzero [Formula: see text] at least one of the extensions [Formula: see text] and [Formula: see text] has no proper subrings. There are weak and strong versions: [Formula: see text] is weakly [Formula: see text]-valuative if for each pair [Formula: see text] such that [Formula: see text], at least one of [Formula: see text] and [Formula: see text] has no proper intermediate rings; [Formula: see text] is strongly [Formula: see text]-valuative if for each nonzero [Formula: see text] at least one of the extensions [Formula: see text] and [Formula: see text] has no proper intermediate rings. Here, we consider the special case that [Formula: see text] is the total quotient ring. Examples are provided to show that even in this setting there are [Formula: see text]-weakly valuative rings that are not [Formula: see text]-valuative, and [Formula: see text]-valuative rings that are not strongly [Formula: see text]-valuative. In contrast, if [Formula: see text] is weakly additively regular, then it is weakly [Formula: see text]-valuative if and only if it is strongly [Formula: see text]-valuative.

Certain towers of ramified minimal ring extensions of commutative rings

ABSTRACTLet p be a prime number. Let B be the idealization ℤ(+)ℤ∕pℤ and N: = pℤ(+)ℤ∕pℤ. Let B⊂R be a ramified (integral minimal) ring extension, with crucial maximal ideal 𝒩 and (necessarily) an element y such that R = B[y], y2∈B, y3∈B and y𝒩⊆𝒩. Then B is the only (commutative unital) ring properly contained between ℤ and R if and only if 𝒩 = N and either or py∉ℤ. Moreover, there are exactly three isomorphism classes of such rings R. Let A: = ℤ∕pαℤ for some integer α≥2. Let A⊂D⊂E be a tower of ramified ring extensions, (necessarily) with an element z such that E = D[z], z2∈D, z3∈D and zℳ⊆ℳ, where ℳ is the crucial maximal ideal of D⊂E. Then D is the only ring properly contained between A and E if and only if either z2∉A or pz∉A. There are at least two, but only finitely many, isomorphism classes of such rings E. We complete the characterization of the rings with exactly two proper (unital) subrings, including a classification up to isomorphism of these rings in the case of characteristic 0.

Pairs of integral domains with most of the intermediate rings PVD

Given a ring extension \(R \subset S\) of integral domains, it is shown that if each proper subring of S containing R is a PVD, then S is a PVD. As an application, we show that if each proper subring of S containing R is a valuation domain (resp., a DVR), then S is valuation domain (resp. S is the quotient field of R).

On the Commutative Rings with At Most Two Proper Subrings

The commutative rings with exactly two proper (unital) subrings are characterized. An initial step involves the description of the commutative rings having only one proper subring.