Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $m,n$ fixed positive integers. (i) If ${\left(d[x,y]\right)}^m={[x,y]}_n$ for all $x,y\in I$, then $R$ is commutative. (ii) If $\mathrm{Char}R\ne 2$ and ${[d\left(x\right),d\left(y\right)]}_m={[x,y]}^n$ for all $x,y\in I$, then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.

Let $R$ be a prime ring of char $R\ne 2$ with a nonzero derivation $d$ and let $U$ be its noncentral Lie ideal. If for some fixed integers $n_1\ge 0,n_2\ge 0,n_3\ge 0$, ${\left(u^{n_1}[d\left(u\right),u]u^{n_2}\right)}^{n_3}\in Z\left(R\right)$ for all $u\in U$, then $R$ satisfies $S_4$, the standard identity in four variables.

Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support $\tau$-tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given $\tau$-tilting module.

Let Λ be an artin algebra. We prove that for each sequence ${\left(h_i\right)}_{i\in \mathbb{Z}}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X{\in }^b\left(\Lambda \right)$, the bounded derived category of Λ, with $lengt{h}_{E\left(X\right)}{H}^i\left(X\right)=h_i$ for all i ∈ ℤ and E(X) the endomorphism ring of X in ${}^b\left(\Lambda \right)$ if and only if ${}^b\left(Mod\Lambda \right)$, the bounded derived category of the category $Mod\Lambda$ of all left Λ-modules, has no generic objects in the sense of [4].

We complete the derived equivalence classification of all weakly symmetric algebras of domestic type over an algebraically closed field, by solving the problem of distinguishing standard and nonstandard algebras up to stable equivalence, and hence derived equivalence. As a consequence, a complete stable equivalence classification of weakly symmetric algebras of domestic type is obtained.

We construct derived equivalences between generalized matrix algebras. We record several corollaries. In particular, we show that the $n$-replicated algebras of two derived equivalent, finite-dimensional algebras are also derived equivalent.

In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in M_n\left(I\right)$, there exist left invertible $U_1,U_2\in M_n\left(R\right)$ and right invertible $V_1,V_2\in M_n\left(R\right)$ such that $U_1{V}_{1}A{U}_2{V}_2=diag(e_1,\cdots ,e_n)$ for idempotents $e_1,\cdots ,e_n\in I$.