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Alborz Azarang

Publications and source records attributed to Alborz Azarang.

11 recordsLinked to original sources
arXiv2026

Similar submodules of projective modules

We introduce a similarity relation between submodules of a module $M$ over a ring $R$, extending the classical notion of similarity for right ideals. Focusing on (faithfully) projective modules, we establish a sharp lower bound for the number of maximal submodules: if $N$ is a maximal submodule of $M$, then either $N$ is fully invariant or $N$ is similar to at least $1+|S|$ distinct maximal submodules, where $S$ is the eigenring of $N$; in particular, $|{\rm Max}(M)|\geq 1+|S|\geq 3$ in the latter case. For projective modules, we construct a canonical one-to-one map from ${\rm Max}(M)$ into ${\rm Max}_r({\rm End}_R(M))$. When $M$ is faithfully projective and ${\rm End}_R(M)$ is right Artinian, we prove that $M$ has finite length and decomposes into a direct sum of local summands. Conversely, if $M$ is a projective right $R$-module with finite length, then $E_E$ has finite length with $\ell(E_E)\leq \ell(M_R)$; moreover, if $M$ is a faithfully projective $R$-module, then $\ell(E_E)=\ell(M_R)$; conversely, if $\ell(E_E)=\ell(M_R)$ holds, then $M$ is slightly compressible. These results are applied to obtain lower bounds on the number of maximal one-sided ideals that are not two-sided, with explicit consequences for matrix rings over infinite algebras.

math.RA↗
arXiv2026

Non-commutative rings with infinitely many maximal subrings

We study rings with infinitely (only finitely) many maximal subrings. We prove that if $M$ is a maximal left/right ideal of a ring $T$ which is not an ideal of $T$, and $R$ is the idealizer of $M$, then $T$ has at least $|R/M|+1$ maximal left/right ideals which are not an ideal of $T$; in particular $T$ has at least $|R/M|+1$ distinct maximal subrings. Moreover, if $T$ is a $K$-algebra over an infinite field $K$, then either $T$ has infinitely many maximal subrings or $T$ is a quasi duo ring with certain algebraic properties similar to commutative rings. We prove that for a simple ring $R$, the ring $R\times R$ has only finitely many maximal subrings if and only if $R$ is finite. Also we study rings which are integral over their centers and have only finitely many maximal subrings. We prove that if $T$ is integral over its center and $T$ has more than $2^{\aleph_0}$ maximal (left/right) ideals, then $T$ has infinitely many maximal subrings. In particular, we see that if a $J$-semisimple ring $T$ is integral over its center and has only finitely many maximal subrings, then $T$ embeds in $S\times \prod_{i\in I}E_i$, where each $E_i$ is an absolutely algebraic field and $S$ is a finite semisimple ring. We see that if $T$ is a left Noetherian algebraic $K$-algebra over an infinite field $K$ and $T$ has only finitely many maximal subrings, then $T$ is a countable left Artinian ring which is integral over $\mathbb{Z}_p$, where $p=Char(K)$. We exactly determine when $T=\prod_{i\in I}\mathbb{M}_{n_i}(E_i)$, where each $E_i$ is a field and $n_i\in\mathbb{N}$, has only finitely many maximal subrings. We see that if $R$ is an infinite Artinian ring, then $\mathbb{M}_n(R)$, $n>1$, and $R\times R$ have infinitely many maximal subrings.

math.RA↗
arXiv2024

Maximal subrings of division rings

The structure and the existence of maximal subrings in division rings are investigated. We see that if $R$ is a maximal subring of a division ring $D$ with center $F$ and $N(R)\neq U(R)\cup \{0\}$, where $N(R)$ is the normalizer of $R$ in $D$, then either $R$ is a division ring with $[D:R]_l=[D:R]_r$ is finite or $R$ is an Ore $G$-domain with certain properties. In particular, if $F\subsetneq C_D(R)$, the centralizer of $R$ in $D$, then $R=C_D(\beta)$ is a division ring, for each $\beta\in C_R(R)\setminus F$, $[D:R]_l$ is finite if and only if $\beta$ is algebraic over $F$, $[D:R]_l=[D:R]_r=[F[\beta]:F]$ and $C_R(R)=F[\beta]$. On the other hand if $R$ does not contains $F$, then $R\cap F=C_R(R)$ is a maximal subring of $F$. Consequently, if a division ring $D$ has a noncentral element which is algebraic over the center of $D$, then $D$ has a maximal subring. In particular, we prove that if $D$ is a non-commutative division ring with center $F$, then either $D$ has a maximal subring or $dim_F(D)\geq |F|$. We study when a maximal subring of a division ring is a left duo ring or certain valuation rings. Finally, we prove that if $D$ is an existentially complete division ring over a field $K$, then $D$ has a maximal subring of the form $C_D(x)$ where $D$ is finite over it. Moreover, if $R$ is a maximal subring of $D$ with $K\subsetneq C_R(R)$, then $R=C_D(x)$ for some $x\in D\setminus K$, which is algebraic over $K$.

math.RA↗
arXiv2024

Maximal subrings of certain non-commutative rings

The existence of maximal subrings in certain non-commutative rings, especially in rings which are integral over their centers, are investigated. We prove that if a ring $T$ is integral over its center, then either $T$ has a maximal subring or $T/J(T)$ is a commutative Hilbert ring with $|Max(T)|\leq 2^{\aleph_0}$ and $|T/J(T)|\leq 2^{2^{\aleph_0}}$. We observe that if $T$ is an algebraic $K$-algebra over a field $K$, then either $T$ has a maximal subring or $U(T)$ is integral over the prime subring of $T$. If $T$ is a left Artinian ring which is integral over its center, then we prove that either $T$ has a maximal subring or $T$ is countable and is integral over its prime subring. We see that if $T$ is a left Noetherian ring which is integral over its center, then either $T$ has a maximal subring or $|T|\leq 2^{\aleph_0}$. We prove that if $T$ is a domain which is integral over its center $C$ and $J(C)=0$, then either $T$ has a maximal subring or $T$ is an integral domain. If $T$ is a reduced ring which is integral over its center and the center of $T$ is a Hilbert ring, then we show that either $T$ has a maximal subring or $T$ is commutative. We see that if a ring $T$ is integral over its center and $R$ is a subring of $T$ with $J(T)\cap R\subseteq J(R)$, then either $T$ has a maximal subring or $J(R)=J(T)\cap R$ and $U(R)=U(T)\cap R$. Finally, we prove that if $T$ is direct product of an infinite family of rings $\{T_i\}_{i\in I}$ and each $T_i$ is integral over its center, then $T$ has a maximal subrings.

math.RA↗
arXiv2024

The conductor ideals of maximal subrings in non-commutative rings

Let $R$ be a maximal subring of a ring $T$, and $(R:T)$, $(R:T)_l$ and $(R:T)_r$ denote the greatest ideal, left ideal and right ideal of $T$ which are contained in $R$, respectively. It is shown that $(R:T)_l$ and $(R:T)_r$ are prime ideals of $R$ and $|Min_R((R:T))|\leq 2$. We prove that if $T_R$ has a maximal submodule, then $(R:T)_l$ is a right primitive ideal of $R$. We investigate that when $(R:T)_r$ is a completely prime (right) ideal of $R$ or $T$. If $R$ is integrally closed in $T$, then $(R:T)_l$ and $(R:T)_r$ are prime one-sided ideals of $T$. We observe that if $(R:T)_lT=T$, then $T$ is a finitely generated left $R$-module and $(R:T)_l$ is a finitely generated right $R$-module. We prove that $Char(R/(R:T)_l)=Char(R/(R:T)_r)$, and if $Char(T)$ is neither zero or a prime number, then $(R:T)\neq 0$. If $|Min(R)|\geq 3$, then $(R:T)$ and $(R:T)_l(R:T)_r$ are nonzero ideals. Finally we study the Noetherian and the Artinian properties between $R$ and $T$.

math.RA↗
arXiv2024

Classical ideals theory of maximal subrings in non-commutative rings

Let $R$ be a maximal subring of a ring $T$. In this paper we study relation between some important ideals in the ring extension $R\subseteq T$. In fact, we would like to find some relation between $Nil_*(R)$ and $Nil_*(T)$, $Nil^*(R)$ and $Nil^*(T)$, $J(R)$ and $J(T)$, $Soc({}_RR)$ and $Soc({}_RT)$, and finally $Z({}_RR)$ and $Z({}_RT)$; especially, in certain cases, for example when $T$ is a reduced ring, $R$ (or $T$) is a left Artinian ring, or $R$ is a certain maximal subring of $T$. We show that either $Soc({}_RR)=Soc({}_RT)$ or $(R:T)_r$ (the greatest right ideal of $T$ which is contained in $R$) is a left primitive ideal of $R$. We prove that if $T$ is a reduced ring, then either $Z({}_RT)=0$ or $Z({}_RT)$ is a minimal ideal of $T$, $T=R\oplus Z({}_RT)$, and $(R:T)=(R:T)_l=(R:T)_r=ann_R(Z({}_RT))$. If $T=R\oplus I$, where $I$ is an ideal of $T$, then we completely determine relation between Jacobson radicals, lower nilradicals, upper nilradicals, socle and singular ideals of $R$ and $T$. Finally, we study the relation between previous ideals of $R$ and $T$ when either $R$ or $T$ is a left Artinian ring.

math.RA↗
arXiv2023

Maximal subrings up to isomorphism of fields

In this paper we study maximal subrings up to isomorphism of fields. It is shown that each field with zero characteristic has infinitely many maximal subrings up to isomorphism. If $K$ is an algebraically closed field and $x$ is an indeterminate over $K$, then we prove that integrally closed maximal subrings of $K(x)$ which contains $K$ are all isomorphic. In particular, if $K$ is an absolutely algebraic field, then $K(x)$ has only finitely many integrally closed maximal subrings up to isomorphism if and only if $K$ is algebraically closed. Also, we show that if $K$ is an absolutely algebraic field then $K$ has only finitely many maximal subrings up to isomorphism if and only if $K$ has only finitely many maximal subrings. We prove that if a commutative ring $R$ with zero characteristic has only finitely many maximal subrings up to isomorphism, then $U(R)\cap\mathbb{P}$ is finite, where $\mathbb{P}$ is the set of natural prime numbers. In particular, if $R$ is a commutative ring with zero characteristic and $Char(R/J(R))\neq 0$, then $R$ has infinitely many maximal subrings up to isomorphism. Maximal subrings up to isomorphism of $K[x]$, $K\times K$ and $K[x]/(x^2)$ for a field $K$ are investigated. If $R$ is a non-field maximal subring of a field $K$ and $\mathcal{A}$ is the set of all maximal subrings of $K$ which are isomorphic to $R$, then we prove that $\mathcal{A}=\{\sigma(R)\ |\ \sigma\in Aut(K)|\}$, in particular $|\mathcal{A}|\leq |Aut(K)|$. Moreover if $K$ is infinite and $|Aut(K)|<|K|$, then $R$ has at least $|K|$-many integrally closed maximal subrings up to isomorphism.

math.AC↗
arXiv2020

Prime Avoidance Property

Let $R$ be a commutative ring, we say that $\mathcal{A}\subseteq Spec(R)$ has prime avoidance property, if $I\subseteq \bigcup_{P\in\mathcal{A}}P$ for an ideal $I$ of $R$, then there exists $P\in\mathcal{A}$ such that $I\subseteq P$. We exactly determine when $\mathcal{A}\subseteq Spec(R)$ has prime avoidance property. In particular, if $\mathcal{A}$ has prime avoidance property, then $\mathcal{A}$ is compact. For certain classical rings we show the converse holds (such as Bezout rings, $QR$-domains, zero-dimensional rings and $C(X)$). We give an example of a compact set $\mathcal{A}\subseteq Spec(R)$, where $R$ is a Prufer domain, which has not $P.A$-property. Finally, we show that if $V,V_1,\ldots, V_n$ are valuation domains for a field $K$ and $V[x]\nsubseteq \bigcup_{i=1}^n V_i$ for some $x\in K$, then there exists $v\in V$ such that $v+x\notin \bigcup_{i=1}^n V_i$.

math.AC↗
arXiv2020

Conch Maximal Subrings

It is shown that if $R$ is a ring, $p$ a prime element of an integral domain $D\leq R$ with $\bigcap_{n=1}^\infty p^nD=0$ and $p\in U(R)$, then $R$ has a conch maximal subring (see \cite{faith}). We prove that either a ring $R$ has a conch maximal subring or $U(S)=S\cap U(R)$ for each subring $S$ of $R$ (i.e., each subring of $R$ is closed with respect to taking inverse, see \cite{invsub}). In particular, either $R$ has a conch maximal subring or $U(R)$ is integral over the prime subring of $R$. We observe that if $R$ is an integral domain with $|R|=2^{2^{\aleph_0}}$, then either $R$ has a maximal subring or $|Max(R)|=2^{\aleph_0}$, and in particular if in addition $dim(R)=1$, then $R$ has a maximal subring. If $R\subseteq T$ be an integral ring extension, $Q\in Spec(T)$, $P:=Q\cap R$, then we prove that whenever $R$ has a conch maximal subring $S$ with $(S:R)=P$, then $T$ has a conch maximal subring $V$ such that $(V:T)=Q$ and $V\cap R=S$. It is shown that if $K$ is an algebraically closed field which is not algebraic over its prime subring and $R$ is affine ring over $K$, then for each prime ideal $P$ of $R$ with $ht(P)\geq dim(R)-1$, there exists a maximal subring $S$ of $R$ with $(S:R)=P$. If $R$ is a normal affine integral domain over a field $K$, then we prove that $R$ is an integrally closed maximal subring of a ring $T$ if and only if $dim(R)=1$ and in particular in this case $(R:T)=0$.

math.AC↗
arXiv2014

On Fields With Only Finitely Many Maximal Subrings

Fields with only finitely many maximal subrings are completely determined. We show that such fields are certain absolutely algebraic fields and give some characterization of them. In particular, we show that the following conditions are equivalent for a field $E$: 1. $E$ has only finitely many maximal subrings. 2. $E$ has a subfield $F$ which has no maximal subrings and $[E:F]$ is finite. 3. Every descending chain $\cdots\subset R_2\subset R_1\subset R_0=E$ where each $R_i$ is a maximal subring of $R_{i-1}$, $i\geq 1$, is finite. Moreover, if one of the above equivalent conditions holds, then $F$ is unique and contains all subfields of $E$ which have no maximal subrings. Furthermore, all chains in $(3)$ have the same length, $m$ say, and $R_m=F$, where $m$ is the sum of all powers of primes in the factorization of $[E:F]$ into prime numbers.\\ We also determine when certain affine rings have only finitely many maximal subrings. In particular, we prove that if $R=F[α_1,\ldots,α_n]$ is an affine integral domain over a field $F$, then $R$ has only finitely many maximal subrings if and only if $F$ has only finitely many maximal subrings and each $α_i$ is algebraic over $F$, which is similar to the celebrated Zariski's Lemma. Finally, we show that if $R$ is an uncountable PID then $R$ has at least $|R|$-many maximal subrings.

math.AC↗