Alborz, Azarang

Azarang A., Parsa N.

Azarang A.

Alinaghizadeh M., Azarang A.

Azarang A.

It is shown that if R is a ring, p a prime element of an integral domain D≤R with ∩n=1∞pnD=0 and p∈U(R), then R has a conch maximal subring (see [14]). We prove that either a ring R has a conch max...

Azarang A.

In this note we give a simple version (with a simple proof) of Noether’s Normalization Lemma which implies Zariski’s Lemma and Hilbert’s Nullstellensatz (weak form) more naturally.

Azarang A.

Let [Formula: see text] be a commutative ring, we say that [Formula: see text] has prime avoidance property, if [Formula: see text] for an ideal [Formula: see text] of [Formula: see text], then there exists [Formula: see text] such that [Formula: see text]. We exactly determine when [Formula: see text] has prime avoidance property. In particular, if [Formula: see text] has prime avoidance property, then [Formula: see text] is compact. For certain classical rings we show the converse holds (such as Bezout rings, [Formula: see text]-domains, zero-dimensional rings and [Formula: see text]). We give an example of a compact set [Formula: see text], where [Formula: see text] is a Prufer domain, which has not prime avoidance property. Finally, we show that if [Formula: see text] are valuation domains for a field [Formula: see text] and [Formula: see text] for some [Formula: see text], then there exists [Formula: see text] such that [Formula: see text].

Azarang A.

Azarang A., Shahrisvand F.

In this paper, we study rings with only finitely many essential right ideals (right fe-rings for short). We see that these rings have some similar properties of semisimple rings. In fact, we prove ...

Azarang A.

Let R be a commutative ring and X = RgMax(R) be the set of all maximal subrings of R. We give a topology on X by putting 𝕏(S) = {T ∈ X | S ⊆ T}, where S ranges over all subrings of R, as a subbase for closed subsets for X. We investigate the decomposition into irreducible components for this topology. It is shown that valuation domains behave similar to prime ideals in Zariski topology in our topology. Further we present an analogous form of the Prime Avoidance Lemma for valuation domains instead of prime ideals. The compactness of 𝕏(S) for certain subrings S of R is determined. Moreover, we characterize fields E for which the space X = RgMax(E) is compact.

Azarang A.

In this note, we generalize the results of [Submaximal integral domains, Taiwanese J. Math. 17(4) (2013) 1395–1412; Which fields have no maximal subrings? Rend. Sem. Mat. Univ. Padova 126 (2011) 213–228; On the existence of maximal subrings in commutative artinian rings, J. Algebra Appl. 9(5) (2010) 771–778; On maximal subrings of commutative rings, Algebra Colloq. 19(Spec 1) (2012) 1125–1138] for the existence of maximal subrings in a commutative noetherian ring. First, we show that for determining when an infinite noetherian ring R has a maximal subring, it suffices to assume that R is an integral domain with |R/I| < |R| for each nonzero ideal I of R. We determine when the latter integral domains have maximal subrings. In particular, we show that every uncountable noetherian ring has a maximal subring.

Azarang A., Oman G.

It is shown that RgMax (R) is infinite for certain commutative rings, where RgMax (R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then | RgMax (R)| ≥ | Irr (D) ∩ U(R)|, where Irr (D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or | RgMax (R)| ≥ ℵ0. It is proved that if R is a zero-dimensional (or semilocal) ring with | RgMax (R)| < ℵ0, then R has nonzero characteristic, say n, and R is integral over ℤn. In particular, it is shown that if R is an uncountable artinian ring, then | RgMax (R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2ℵ0, then | RgMax (R)| ≥ 2ℵ0. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R2 ⊂ R1 ⊂ R0 = R where each Ri is a maximal subring of Ri-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

Azarang A., Karamzadeh O.A., Namazi A.

We study the existence of maximal subrings and hereditary properties between a ring and its maximal subrings. Some new techniques for establishing the existence of maximal subrings are presented. It is shown that if R is an integral domain and S is a maximal subring of R, then the relation dim(R) = 1 implies that dim(S) = 1 and vice versa if and only if (S : R) = 0. Thus, it is shown that if S is a maximal subring of a Dedekind domain R integrally closed in R; then S is a Dedekind domain if and only if S is Noetherian and (S : R) = 0. We also give some properties of maximal subrings of one-dimensional valuation domains and zero-dimensional rings. Some other hereditary properties, such as semiprimarity, semisimplicity, and regularity are also studied.

Azarang A.

Azarang A., Karamzadeh O.A.

Ouzzaouit O., Tamoussit A.

A ring R is called an FMR-ring if it is finite modulo all its maximal ideals. In this paper, we investigate some properties of FMR-rings in various ring extensions. Mainly we are concerned with the charachterization of FMR-rings arising from bi-amalgamated algebras and trivial ring extensions.

Alinaghizadeh M., Azarang A.

Chaturvedi A.K., Kumar N.

O. A. S. Karamzadeh et al. [On rings with a unique proper essential right ideal, Fund. Math. 183 (2004) 229–244] introduced a class of modules with unique proper essential submodule, such modules are called as ue-modules. A. Azarang and F. Shahrisvand [Rings with only finitely many essential right ideals, Comm. Algebra 47 (2019) 2843–2854] introduced the idea of fe-module and called a module with finitely many essential submodules as a fe-module. Here, we introduce and study the classes of modules dual to them. The class of modules with finitely many small submodules is defined as fs-modules and modules with a unique nonzero small submodule are defined as us-modules.

Azarang A.

It is shown that if R is a ring, p a prime element of an integral domain D≤R with ∩n=1∞pnD=0 and p∈U(R), then R has a conch maximal subring (see [14]). We prove that either a ring R has a conch max...

Koç S.

The purpose of the paper is to introduce and study weakly 2-prime ideals in commutative rings. Let A be a commutative ring with a nonzero identity. A proper ideal P of A is said to be a weakly 2-pr...

Jarboui N., Al-Kuleab N., Almallah O.

The commutative ring extensions with exactly two non-Artinian intermediate rings are characterized. An initial step involves the description of the commutative ring extensions with only one non-Artinian intermediate ring.

Kumar R., Gaur A.

Let R be a commutative ring with identity. We study the concept of pointwise maximal subrings of a ring. A ring R is called a pointwise maximal subring of a ring T if $$R\subset T$$ and for each $$t\in T{\setminus } R$$ , the ring extension $$R[t]\subseteq T$$ has no proper intermediate ring. A characterization of local, integrally closed pointwise maximal subrings of a ring is given. Let G be a subgroup of the group of automorphisms of T. Then the integrally closed pointwise maximality is a G-invariant property of ring extension under some conditions. We also discuss the number of overrings and the Krull dimension of pointwise maximal subrings of a ring. The pointwise maximal subrings of the polynomial ring R[X] are also discussed.

Azarang A.

Let [Formula: see text] be a commutative ring, we say that [Formula: see text] has prime avoidance property, if [Formula: see text] for an ideal [Formula: see text] of [Formula: see text], then there exists [Formula: see text] such that [Formula: see text]. We exactly determine when [Formula: see text] has prime avoidance property. In particular, if [Formula: see text] has prime avoidance property, then [Formula: see text] is compact. For certain classical rings we show the converse holds (such as Bezout rings, [Formula: see text]-domains, zero-dimensional rings and [Formula: see text]). We give an example of a compact set [Formula: see text], where [Formula: see text] is a Prufer domain, which has not prime avoidance property. Finally, we show that if [Formula: see text] are valuation domains for a field [Formula: see text] and [Formula: see text] for some [Formula: see text], then there exists [Formula: see text] such that [Formula: see text].

Azarang A.

Gottlieb C.

Karamzadeh O.A., Nazari N.

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exac...

Izelgue L., Ouzzaouit O.

A G–ring is any commutative ring R with a nonzero identity such that the total quotient ring $$\mathbf {T}(R)$$ is finitely generated as a ring over R. A G–ring pair is an extension of commutative rings $$A\hookrightarrow B$$ , such that any intermediate ring $$A\subseteq R\subseteq B$$ is a G–ring. In this paper we investigate the transfer of the G–ring property among pairs of rings sharing an ideal. Our main result is a generalization of a theorem of David Dobbs about G–pairs to rings with zero divisors.

Dobbs D.E.

Let κ be a cardinal number. If κ ≥ 2, then there exists a (commutative unital) ring A such that the set of A-algebra isomorphism classes of minimal ring extensions of A has cardinality κ. The preceding statement fails for κ = 1 and, if A must be nonzero, it also fails for κ = 0. If $$ \kappa \leq \aleph _{0} $$ , then there exists a ring whose set of maximal (unital) subrings has cardinality κ. If an infinite cardinal number κ is of the form κ = 2 λ for some (infinite) cardinal number λ, then there exists a field whose set of maximal subrings has cardinality κ.

Izelgue L., Ouzzaouit O.

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

Nicholas J. Werner

Azarang A.

It is shown that if R is a ring, p a prime element of an integral domain D≤R with ∩n=1∞pnD=0 and p∈U(R), then R has a conch maximal subring (see [14]). We prove that either a ring R has a conch max...

Azarang A.

Gottlieb C.

Azarpanah F., Karamzadeh O.A., Keshtkar Z., Olfati A.R.

Maubach S., Stampfli I.

Let $\textbf{k}$ be an algebraically closed field. We classify all maximal $\textbf{k}$-subalgebras of any one-dimensional finitely generated $\textbf{k}$-domain. In dimension two, we classify all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, t^{-1}, y]$. To the authors' knowledge, this is the first such classification result for an algebra of dimension $> 1$. In the course of this study, we classify also all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that contain a coordinate. Furthermore, we give examples of maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that do not contain a coordinate.

Azarang A.

Let R be a commutative ring and X = RgMax(R) be the set of all maximal subrings of R. We give a topology on X by putting 𝕏(S) = {T ∈ X | S ⊆ T}, where S ranges over all subrings of R, as a subbase for closed subsets for X. We investigate the decomposition into irreducible components for this topology. It is shown that valuation domains behave similar to prime ideals in Zariski topology in our topology. Further we present an analogous form of the Prime Avoidance Lemma for valuation domains instead of prime ideals. The compactness of 𝕏(S) for certain subrings S of R is determined. Moreover, we characterize fields E for which the space X = RgMax(E) is compact.

Azarang A.

In this note, we generalize the results of [Submaximal integral domains, Taiwanese J. Math. 17(4) (2013) 1395–1412; Which fields have no maximal subrings? Rend. Sem. Mat. Univ. Padova 126 (2011) 213–228; On the existence of maximal subrings in commutative artinian rings, J. Algebra Appl. 9(5) (2010) 771–778; On maximal subrings of commutative rings, Algebra Colloq. 19(Spec 1) (2012) 1125–1138] for the existence of maximal subrings in a commutative noetherian ring. First, we show that for determining when an infinite noetherian ring R has a maximal subring, it suffices to assume that R is an integral domain with |R/I| < |R| for each nonzero ideal I of R. We determine when the latter integral domains have maximal subrings. In particular, we show that every uncountable noetherian ring has a maximal subring.

Azarang A., Oman G.

It is shown that RgMax (R) is infinite for certain commutative rings, where RgMax (R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then | RgMax (R)| ≥ | Irr (D) ∩ U(R)|, where Irr (D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or | RgMax (R)| ≥ ℵ0. It is proved that if R is a zero-dimensional (or semilocal) ring with | RgMax (R)| < ℵ0, then R has nonzero characteristic, say n, and R is integral over ℤn. In particular, it is shown that if R is an uncountable artinian ring, then | RgMax (R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2ℵ0, then | RgMax (R)| ≥ 2ℵ0. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R2 ⊂ R1 ⊂ R0 = R where each Ri is a maximal subring of Ri-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

Azarang A., Karamzadeh O.A., Namazi A.

We study the existence of maximal subrings and hereditary properties between a ring and its maximal subrings. Some new techniques for establishing the existence of maximal subrings are presented. It is shown that if R is an integral domain and S is a maximal subring of R, then the relation dim(R) = 1 implies that dim(S) = 1 and vice versa if and only if (S : R) = 0. Thus, it is shown that if S is a maximal subring of a Dedekind domain R integrally closed in R; then S is a Dedekind domain if and only if S is Noetherian and (S : R) = 0. We also give some properties of maximal subrings of one-dimensional valuation domains and zero-dimensional rings. Some other hereditary properties, such as semiprimarity, semisimplicity, and regularity are also studied.

OMAN G.

Let R be an associative ring with identity, and let I be an (left, right, two-sided) ideal of R. Say that I is small if |I| < |R| and large if |R/I| < |R|. In this paper, we present results on small and large ideals. In particular, we study their interdependence and how they influence the structure of R. Conversely, we investigate how the ideal structure of R determines the existence of small and large ideals.

Azarang A.

Ghadermazi M., Karamzadeh O.A., Namdari M.

Let Cc(X) = {f ∈ C(X) : f(X) is countable}. Similar to C(X) it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring Cc(X) is either Cc(X) or a semiprime (resp. prime) ideal in Cc(X). For an ideal I in Cc(X), it is observed that I and √ I have the same largest zc-ideal. If X is any topological space, we show that there is a zero-dimensional space Y such that Cc(X) ∼= Cc(Y ). Consequently, if X has only countable number of components, then Cc(X) ∼= C(Y ) for some zero-dimensional space Y . Spaces X for which Cc(X) is regular (called CP -spaces) are characterized both algebraically and topologically and it is shown that P -spaces and CP -spaces coincide when X is zero-dimensional. In contrast to C∗(X), we observe that Cc(X) enjoys the algebraic properties of regularity, א0selfinjectivity and some others, whenever C(X) has these properties. Finally an example of a space X such that Cc(X) is not isomorphic to any C(Y ) is given.

Hartshorne R.

Azarang A., Karamzadeh O.A.

Dobbs D.E., Picavet G., Picavet-LʼHermitte M.

Several parallel characterizations of the FIP and FCP properties are given. Also, a number of results about FCP are generalized from domains to arbitrary (commutative) rings. Let R ⊆ S be rings, with R ¯ the integral closure of R in S . Then R ⊆ S satisfies FIP (resp., FCP) if and only if both R ⊆ R ¯ and R ¯ ⊆ S satisfy FIP (resp., FCP). If R is integrally closed in S , then R ⊆ S satisfies FIP ⇔ R ⊆ S satisfies FCP ⇔ ( R , S ) is a normal pair such that Supp R ( S / R ) is finite. If R ⊆ S is integral and has conductor C , then R ⊆ S satisfies FCP if and only if S is a finitely generated R -module such that R / C is an Artinian ring. The characterizations of FIP and FCP for integral extensions feature natural roles for the intermediate rings arising from seminormalization and t-closure.