Formal Triangular Matrix Rings Research Articles

Special Properties of Morita Contexts

In this paper we continue the study of various ring theoretic properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch, completely primary, quasi-duo, 2-primal, NI, semiprimitive, projective-free, etc. We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective.

Dual modules, f-projective modules and Π-coherence over trivial ring extensions

Let R ⋉ M be a trivial extension of a ring R by an R-R-bimodule M. We first describe explicitly the structure of a dual module over R ⋉ M . Then we characterize Mittag-Leffler modules and f-projective modules over R × M. Finally, we investigate when R ⋉ M is a left Π-coherent ring. In addition, some applications are given in formal triangular matrix rings.

Semiregular, semiperfect and semipotent matrix rings relative to an ideal

This paper investigates relative ring theoretical properties in the context of formal triangular matrix rings. The first aim is to study the semiregularity of formal triangular matrix rings relative to an ideal. We prove that the formal triangular matrix ring $T$ is $T'$-semiregular if and only if $A$ is $I$-semiregular, $B$ is $K$-semiregular and $N=M$ for an ideal $T'=\bigl(\begin{smallmatrix} I & 0\\ N & K \end{smallmatrix}\bigr)$ of $T=\bigl(\begin{smallmatrix} A & 0\\ M & B \end{smallmatrix}\bigr).$ We also discuss the relative semiperfect formal triangular matrix rings in relation to the strong lifting property of ideals. Moreover, we have considered the behavior of relative semipotent and potent property of formal triangular matrix rings. Several examples are provided throughout the paper in order to highlight our results.

Formal matrix rings: Automorphisms

Automorphism groups of formal matrix algebras with zero trace ideals are studied. Such an algebra is represented as a splitting extension of some ring by some nilpotent ideal. Using this extension, the study of the structure of the automorphism group of an algebra in a certain sense is reduced to the study of the structure of some its subgroups and quotient groups. Then the structure of these subgroups and factor groups is found. The case of formal triangular matrix rings is specially considered. At the end of the paper we study automorphisms of ordinary matrix rings. In this case, a well-known homomorphism from the group of outer automorphisms to the Picard group of a certain ring is used.

The Construction of Armendariz Ring using Formal Triangle Matrix Ring

Trinion and Quaternion numbers are one of the hypercomplex numbers which is an extensions of the complex number. From Trinion and Quaternion numbers, a bimodule can be formed which is an ordered pair of Trinion and Quaternion. Furthermore, Trinion number, Quaternion number, and their bimodule can be formed into a Formal Triangle Matrix. The Formal Triangle Matrix is better known as the Upper Triangle Matrix. Since Trinion number, Quaternion number and their bimodule are rings, then the Formal Triangle Matrix can be called as the Formal Triangular Matrix Ring. The purpose of this study is to construct the Armendariz Ring using the Formal Triangular Matrix Ring. The obtained results will show that the Formal Triangular Matrix Rings are the -Skew Armendariz Ring and the -Skew -Armendariz Ring, where is a Ring Endomorphism and is -derivation.

Ding modules and dimensions over formal triangular matrix rings

Let T=\bigl(\begin{smallmatrix}A&0\\U&B\end{smallmatrix}\bigr) be a formal triangular matrix ring, where A and B are rings and U is a (B, A) -bimodule. We prove: (1) If U_{A} and _{B}U have finite flat dimensions, then a left T -module \bigl(\begin{smallmatrix}M_1\\ M_2\end{smallmatrix}\bigr)_{\varphi^{M}} is Ding projective if and only if M_1 and M_2/{\operatorname{im}(\varphi^{M})} are Ding projective and the morphism \varphi^{M} is a monomorphism. (2) If T is a right coherent ring, _{B}U has finite flat dimension, U_{A} is finitely presented and has finite projective or \operatorname{FP} -injective dimension, then a right T -module (W_1, W_2)_{\varphi_{W}} is Ding injective if and only if W_1 and \ker(\widetilde{\varphi_W}) are Ding injective and the morphism \widetilde{\varphi_W} is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a T -module.

Homological aspects of formal triangular matrix rings

Let $T=\biggl(\begin{matrix} A&0\\U&B\end{matrix}\biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We first give some computing formulas of projective, injective, flat and $FP$-injective dimensions of special left $T$-modules. Then we establish some formulas of (weak) global dimensions of $T$. It is proven that (1) If $U_{A}$ is flat and $_{B}U$ is projective, $lD(A)\neq lD(B)$, then $lD(T)={\rm max}\{lD(A),lD(B)\}$; (2) If $U_{A}$ and $_{B}U$ are flat, $wD(A)\neq wD(B)$, then $wD(T)={\rm max}\{wD(A),wD(B)\}$.

Silting and cosilting modules over trivial ring extensions

We establish connections between (co, weak) silting modules over a ring R and (co, weak) silting modules over a trivial extension of R. As applications, we describe how to construct (partial) tilting, cotilting and Tor-tilting modules over trivial ring extensions. In addition, we characterize (partial) silting, cosilting and weak silting modules over formal triangular matrix rings.

Gorenstein AC-Projective and AC-Injective Modules over Formal Triangular Matrix Rings

Let [Formula: see text] and [Formula: see text] be rings and [Formula: see text] a [Formula: see text]-bimodule. If [Formula: see text] is flat and [Formula: see text] is finitely generated projective (resp., [Formula: see text] is finitely generated projective and [Formula: see text] is flat), then the characterizations of level modules and Gorenstein AC-projective modules (resp., absolutely clean modules and Gorenstein AC-injective modules) over the formal triangular matrix ring [Formula: see text] are given. As applications, it is proved that every Gorenstein AC-projective left [Formula: see text]-module is projective if and only if each Gorenstein AC-projective left [Formula: see text]-module and [Formula: see text]-module is projective, and every Gorenstein AC-injective left [Formula: see text]-module is injective if and only if each Gorenstein AC-injective left [Formula: see text]-module and [Formula: see text]-module is injective. Moreover, Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring [Formula: see text] are studied.

Homological dimensions of special modules over formal triangular matrix rings

Let [Formula: see text] be a formal triangular matrix ring, where [Formula: see text] and [Formula: see text] are rings and [Formula: see text] is a [Formula: see text]-bimodule. We give some computing formulas of homological dimensions of special [Formula: see text]-modules. As an application, we describe the structures of [Formula: see text]-tilting left [Formula: see text]-modules.

Lifting properties of formal triangular matrix rings

Lifting properties of formal triangular matrix rings

The structures of dual modules over formal triangular matrix rings

The structures of dual modules over formal triangular matrix rings

On Some n-C2 and Strongly C2 Extensions

Let R be a ring and n be a positive integer. Then R is called a left n-C2-ring (strongly left C2-ring) if every n-generated (finitely generated) proper right ideal of R has nonzero left annihilator. We discuss some n-C2 and strongly C2 extensions, such as trivial extensions, formal triangular matrix rings, group rings and [Formula: see text][D, C].

Duality pairs and FP-injective modules over formal triangular matrix rings

Suppose that is a formal triangular matrix ring, where A and B are rings and U is a (B, A)-bimodule. We first study how to construct duality pairs of T-modules using duality pairs of A-modules and B-modules. Then we describe the structures of FP-injective left T-modules. Some applications are also given.

ANNIHILATOR-STABILITY AND TWO QUESTIONS OF NICHOLSON

AbstractAn element a in a ring R is left annihilator-stable (or left AS) if, whenever $Ra+{\rm l}(b)=R$ with $b\in R$ , $a-u\in {\rm l}(b)$ for a unit u in R, and the ring R is a left AS ring if each of its elements is left AS. In this paper, we show that the left AS elements in a ring form a multiplicatively closed set, giving an affirmative answer to a question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.]. This result is used to obtain a necessary and sufficient condition for a formal triangular matrix ring to be left AS. As an application, we provide examples of left AS rings R over which the triangular matrix rings ${\mathbb T}_n(R)$ are not left AS for all $n\ge 2$ . These examples give a negative answer to another question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.] whether R/J(R) being left AS implies that R is left AS.

Cotorsion pairs and approximation classes over formal triangular matrix rings

Let T=(A0UB) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. Let C1 and C2 be two classes of left A-modules, D1 and D2 be two classes of left B-modules, we prove that: (1) If ToriA(U,C1)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (PD1C1,AD2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. (2) If ExtBi(U,D2)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (AD1C1,ID2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. In addition, we characterize some special preenveloping classes and precovering classes in T-Mod.

Gorenstein flat modules and dimensions over formal triangular matrix rings

Let T=(A0UB) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. We prove that, if T is a right coherent ring, UB has finite flat dimension, UA has finite flat or injective dimension, then a left T-module (M1M2)φM is Gorenstein flat if and only if M1 is a Gorenstein flat left A-module, M2/im(φM) is a Gorenstein flat left B-module and the morphism φM:U⊗AM1→M2 is a monomorphism. This result extends an earlier result in this direction. In addition, we give an estimate of Gorenstein flat dimension of a left T-module.

Associated primes and primary right ideals of generalized triangular matrix rings

Let be an (S, R)-bimodule of the rings R and S. We determine the associated primes of a formal triangular matrix ring Indeed, we show that We then obtain necessary and sufficient conditions for the tertiary decomposition theory to exist on a module over an arbitrary ring. Consequently, we classify all the tertiary right ideals of the formal triangular matrix rings.

Rings in Which Nilpotent Elements Belong to Socle

Prompted by the notion of J-reduced rings in which the nilpotent elements belong to the Jacobson radical, we introduce right (left) \({\mathfrak {S}}\)-reduced rings as those satisfying the condition that the nilpotent elements reside in the right (left) socle. It turns out that a ring R is right (left) \({\mathfrak {S}}\)-reduced if and only if the quotient ring of R modulo its right (left) socle is reduced. Among other things, we shall determine exactly when the idealization of a module over a commutative ring as well as the formal triangular matrix rings has this property.

The set of singular elements of a ring with respect to a module

Let [Formula: see text] be a ring and [Formula: see text] be a right [Formula: see text]-module. In this paper, we introduce the set [Formula: see text] for some essential submodule [Formula: see text] of [Formula: see text] of singular elements of[Formula: see text] with respect to[Formula: see text] , and we investigate the properties of it. For example, it is shown that [Formula: see text] is an ideal of [Formula: see text] and [Formula: see text]. Also if [Formula: see text] is a semiprime right Goldie ring, then [Formula: see text], where [Formula: see text] is the right singular ideal of [Formula: see text]. We prove that if [Formula: see text] is a semisimple module or a prime module, then [Formula: see text]. For any submodule [Formula: see text] of [Formula: see text], we have [Formula: see text] and if [Formula: see text], then [Formula: see text]. We show that [Formula: see text] and [Formula: see text]. In the end, the singular elements of some rings with respect to the formal triangular matrix ring are investigated.