In the present paper we describe Jordan matrix algebras over a field by generators and relations. We prove that the minimun number of generators of some special Jordan matrix algebras over a field is $2$.

Let $\mathcal{A}$ be a unital algebra with nontrivial idempotents. We considered $\mathcal{A}$ as a superalgebra according to Ghahramani and Zadeh's method. We provided a description of supercommuting maps on $\mathcal{A}$. As a consequence, we gave a description of supercommuting maps on matrix algebras, which is different from the result on commuting maps of matrix algebras. Finally, we proved that every supercommuting map on triangular algebras is a commuting map.

Let $D$ be a division ring such that either char $(D)\ne 2,3$ or $D$ is not a field and char $(D)\ne 2$. Let $R=M_n(D)$ be the matrix ring over $D$, where $n>1$. Let $m,k$ be fixed invertible elements in $R$. The main purpose of the paper is to give a description of a bijective additive map $f$: $R\to R$, satisfying the identity $f(x)f(y)=m$ for every $x,y\in R$ with $xy=k$, which gives a correct version of a result due to Catalano et al. in 2019.