‎let $mathcal a$ and $mathcal b$ be unital rings‎, ‎and $mathcal m$‎ ‎be an $(mathcal a‎, ‎mathcal b)$-bimodule‎, ‎which is faithful as a‎ ‎left $mathcal a$-module and also as a right $mathcal b$-module‎. ‎let ${mathcal u}=mbox{rm tri}(mathcal a‎, ‎mathcal m‎, ‎mathcal‎ ‎b)$ be the triangular ring and ${mathcal z}({mathcal u})$ its‎ ‎center‎. ‎assume that $f:{mathcal u}rightarrow{mathcal u}$ is...

Let $T=\bigl(\begin{smallmatrix}A&0\U\&B\end{smallmatrix}\bigr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings $U$ is $(B, A)$-bimodule. We prove: (1) If $U\_{A}$ ${B}U$ have finite flat dimensions, then left $T$-module $\bigl(\begin{smallmatrix}M\_1\ M\_2\end{smallmatrix}\bigr){\varphi^{M}}$ Ding projective if only $M\_1$ $M\_2/{\operatorname{im}(\varphi^{M})}$ the morphism $...

in this paper, some elementary operations on triangular fuzzynumbers (tfns) are defined. we also define some operations on triangularfuzzy matrices (tfms) such as trace and triangular fuzzy determinant(tfd). using elementary operations, some important properties of tfms arepresented. the concept of adjoints on tfm is discussed and some of theirproperties are. some special types of tfms (e.g. pu...

In this paper, we introduce the concept of the generalized AIP-rings as a generalization of the generalized quasiBaer rings and generalized p.p.-rings. We show that the class of the generalized AIP-rings is closed under direct products and Morita invariance. We also characterize the 2-by-2 formal upper triangular matrix rings of this new class of rings. Finally, we provide several examples to s...

Let U be a (B, A)-bimodule, A and B rings, formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using left global dimensions B, estimate dimension T-module.

‎Let $mathcal A$ and $mathcal B$ be unital rings‎, ‎and $mathcal M$‎ ‎be an $(mathcal A‎, ‎mathcal B)$-bimodule‎, ‎which is faithful as a‎ ‎left $mathcal A$-module and also as a right $mathcal B$-module‎. ‎Let ${mathcal U}=mbox{rm Tri}(mathcal A‎, ‎mathcal M‎, ‎mathcal‎ ‎B)$ be the triangular ring and ${mathcal Z}({mathcal U})$ its‎ ‎center‎. ‎Assume that $f:{mathcal U}rightarrow{mathcal U}$ is...