begin{abstract} If $F,D:Rto R$ are additive mappings which satisfy $F(x^{n}y^{n})=x^nF(y^{n})+y^nD(x^{n})$ for all $x,yin R$. Then, $F$ is a generalized left derivation with associated Jordan left derivation $D$ on $R$. Similar type of result has been done for the other identity forcing to generalized derivation and at last an example has given in support of the theorems. end{abstract}

We introduced a new nonassociative algebra, namely, left almost algebra, and discussed some of its genetic properties. We discussed the relation of this algebra with flexible algebra, Jordan algebra, and generalized Jordan algebra.

In this paper, left derivations and Jordan left derivations in full and upper triangular matrix rings over unital associative rings are characterized.

In this paper, left derivations and Jordan left derivations in full and upper triangular matrix rings over unital associative rings are characterized.

We show that the \(\alpha\)-fractional bilinear indicator/cube testing constant \(\mathcal{BICT}_{T^{\alpha }}\left( \sigma ,\omega \right) \equiv \sup_{Q\in \mathcal{P}^{n}}\sup_{E,F\subset Q}\frac{1}{\sqrt{\left\vert Q\right\vert_{\sigma }\left\vert Q\right\vert _{\omega }}}\left\vert \int_{F}T_{\sigma}^{\alpha }\left( \mathbf{1}_{E}\right) \omega \right\vert ,\) defined for any singular in...

We determine conditions under which a left Jordan derivation defined on an $MA$-semiring $S$ is this semiring and prove when implies the commutativity of $S$.