Narrow Results

The complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of Abelian groups. We provide a new polynomial time algorithm for deciding the equation solvability problem over certain semidirect products, where the first factor is not necessarily Abelian. Our main idea is to represent such groups as matrix groups, and reduce the original problem to equation solvability over the underlying field. Further, we apply this new method to give a much more efficient algorithm for equation solvability over nilpotent rings than previously existed.

In a matrix ring R = 𝕄₂(S) where S is a commutative ring, we study equations of the form XY - YX = U ∈ GL₂(S), focusing on matrices in R that can appear as X or as XY in such equations. These are the completable and the reflectable matrices in R. For matrices A ∈ R with a zero row or with a constant diagonal, explicit and "computer-checkable" criteria are found for A to be completable or reflectable. A formula for det (XY - YX) discovered recently with Shomron connects this study to diophantine questions about the representation of units of the ground ring S by quadratic forms of the type px² + qy².

Generalizing the notion of nil-cleanness from [A. J. Diesl, Nil clean rings, J. Algebra383 (2013) 197–211], in parallel to [P. V. Danchev and W. Wm. McGovern, Commutative weakly nil clean unital rings, J. Algebra425 (2015) 410–422], we define the concept of weak nil-cleanness for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean.

A nonzero ring is said to be fine if every nonzero element in it is a sum of a unit and a nilpotent element. We show that fine rings form a proper class of simple rings, and they include properly the class of all simple artinian rings. One of the main results in this paper is that matrix rings over fine rings are always fine rings. This implies, in particular, that any nonzero (square) matrix over a division ring is the sum of an invertible matrix and a nilpotent matrix.

A nonzero ring is called a UN-ring if every nonunit is a product of a unit and a nilpotent element. We show that all simple Artinian rings are UN-rings and that the UN-rings whose identity is a sum of two units (e.g. if 2 is a unit), form a proper class of 2-good rings (in the sense of P. Vámos). Thus, any noninvertible matrix over a division ring is the product of an invertible matrix and a nilpotent matrix.

A ring $R$ is defined to be 2-nil-good if every element in $R$ is the sum of two units and a nilpotent. Fundamental properties of such rings are obtained. We prove that every strongly $\pi$-regular ring is 2-nil-good if and only if the identity is the sum of two units. One of the main result of this paper is that every square matrix ring over J-fine rings is 2-nil-good. We establish 2-nil-good property for Morita contexts. This implies, in particular, that every matrix ring over 2-nil-good rings is 2-nil-good. Furthermore, we prove that the ring ${𝕃𝕋𝔻𝔽𝕄}_{\mathbb{N}}(R)$ of all lower triangular diagonal-finite matrices over a 2-good ring $R$ is 2-nil-good.

In this paper, we state a generalization of the ring of integer-valued polynomials over upper triangular matrix rings. The set of integer-valued polynomials over some block matrix rings is studied. In fact, we consider the set of integer-valued polynomials $\mathrm{\text{Int}}{(}_s{L}_n(D))=\{f\in {}_s{L}_n(K)[x]|f{(}_s{L}_n(D)){\subseteq }_s{L}_n(D)\}$ for each $1\le s\le n-1$, where $D$ is an integral domain with quotient field $K$ and ${}_s{L}_n(D)$ is a block matrix ring between upper triangular matrix ring $T_n(D)$ and full matrix ring $M_n(D)$. In fact, we have ${}_{n-1}{L}_n(D)=T_n(D)$. It is known that the sets of integer-valued polynomials $\mathrm{\text{Int}}(T_n(D))=\{f\in T_n(K)[x]|f(T_n(D))\subseteq T_n(D)\}$ and $\mathrm{\text{Int}}(M_n(D))=\{f\in M_n(K)[x]|f(M_n(D))\subseteq M_n(D)\}$ are rings. We state some relations between the rings $\mathrm{\text{Int}}(T_s(D)),\mathrm{\text{Int}}(M_{n-s}(D))$ and the partitions of $\mathrm{\text{Int}}{(}_s{L}_n(D))$. Then, we show that the set $\mathrm{\text{Int}}{(}_s{L}_n(D))$ is a ring for each $1\le s\le n-1$. Further, it is proved that if the ring $\mathrm{\text{Int}}(M_{n-s}(D))$ is not Noetherian then the ring $\mathrm{\text{Int}}{(}_s{L}_n(D))$ is not Noetherian, too. Finally, some properties and relations are stated between the rings $\mathrm{\text{Int}}{(}_s{L}_n(D))$, $\mathrm{\text{Int}}(T_s(D))$ and $\mathrm{\text{Int}}(M_{n-s}(D))$.

We study rings in which $ab$ nonzero idempotent implies $ba$ is also an idempotent. We call such rings i-reversible. Besides studying the basic properties of i-reversible rings, we characterize i-reversible triangular matrix rings, i-reversible matrix rings over commutative rings and i-reversible exchange rings.

A ring $R$ is called a right C-injective ring if every homomorphism from a closed right ideal of $R$ to $R_R$ can be extended to one from $R_R$ to $R_R$. It is clear that a right CS ring must be right C-injective. Left C-injective rings can be defined similarly. Properties of C-injective rings are explored in this paper. It is shown that a left C-injective ring may not be right C-injective and a right C-injective ring may not be right CS. Some extensions of C-injective rings are discussed.

The notion of a convolution type is introduced. Imposing such a type on a ring gives the corresponding convolution ring. Under this umbrella, a wide variety of ring constructions can be covered, including polynomials, matrices, incidence algebras, necklace rings, group rings and quaternion rings. Here the influence of the convolution type on the corresponding convolution ring is investigated, in particular on the existence of homomorphisms and ideals.

Let $R$ be a ring with an endomorphism $\sigma$, $F\cup \{0\}$ the free monoid generated by $U=\{u_1,\dots ,u_t\}$ with 0 added, and $M$ a factor of $F$ obtained by setting certain monomials in $F$ to 0 such that $M^n=0$ for some $n$. Then we can form the non-semiprime skew monoid ring $R[M;\sigma ]$. A local ring $R$ is called bleached if for any $j\in J(R)$ and any $u\in U(R)$, the abelian group endomorphisms $l_u-r_j$ and $l_j-r_u$ of $R$ are surjective. Using $R[M;\sigma ]$, we provide various classes of both bleached and non-bleached local rings. One of the main problems concerning strongly clean rings is to characterize the rings $R$ for which the matrix ring $M_n(R)$ is strongly clean. We investigate the strong cleanness of the full matrix rings over the skew monoid ring $R[M;\sigma ]$.

Let $R$ be a ring and $M$ be a right $R$-module. In this paper, we introduce the set $Z_R(M)=\{r\in R|Nr=0,$ for some essential submodule $N$ of $M\}$ of singular elements of$R$ with respect to$M$, and we investigate the properties of it. For example, it is shown that $Z_R(M)$ is an ideal of $R$ and $Z_R(M)={\bigcup }_{N{\le }_{e}M}{\mathrm{\text{ann}}}_r^R(N)$. Also if $R$ is a semiprime right Goldie ring, then $Z_R(R_R)=Z(R_R)=0$, where $Z(R_R)$ is the right singular ideal of $R$. We prove that if $M$ is a semisimple module or a prime module, then $Z_R(M)={\mathrm{\text{ann}}}_r^R(M)$. For any submodule $N$ of $M$, we have $Z_R(M)\subseteq Z_R(N)$ and if $N{\le }_{e}M$, then $Z_R(M)=Z_R(N)$. We show that $Z_{R[X]}(R[X])=(Z_R(R))[X]$ and $Z_{M_n(R)}(M_n(R))=M_n(Z_R(R))$. In the end, the singular elements of some rings with respect to the formal triangular matrix ring are investigated.