This article is cited in
3 papers
Quasiregularity and primitivity relative to right ideals of a ring
V. A. Andrunakievich,
Yu. M. Ryabukhin
Abstract:
Let
$R$ be an associative ring, and
$P$ a right ideal of
$R$, i.e.
$P\lhd R_R$. An element
$q\in R$ is quasiregular relative to
$P$ if
$q+t-qt\in R$ for a suitable
$t\in R$. A right ideal
$Q$ is quasiregular relative to
$P$ if all the elements of
$Q$ are quasiregular relative to
$P$. If
$M,P\lhd R_R$, then put:
$$
\lambda (M,P)=\{r\in R\mid rP\subseteq M\},\qquad
M:P=\{r\in R\mid Pr\subseteq M\}.
$$
Theorem 1. {\it Let
$P\lhd R_R$. Then the sum
$(R,P)$ of all right ideals which are quasiregular relative to
$P$ is itself quasiregular relative to
$P$. Moreover
$,$ $\mathscr J(R,P)=\bigcap\{M:\lambda (M,P)\mid M$ is a maximal modular right ideal of
$R,M\supseteq P\}$.}
We say that
$P$ is a primitive right ideal of the ring
$R$ if there exists a maximal modular right ideal
$M$ satisfying
$P=M:\lambda(M,P)$. If
$P=0$, then
$0=M:R$, and therefore the ring
$R$ is primitive.
Density theorem. {\it Let
$P$ be a primitive right ideal of the ring
$R,$ and
$M$ any of the maximal modular right ideals corresponding to
$P,$ i.e.
$P=M:\lambda(M,P)$. Consider the irreducible right
$R$-module
$\mathfrak M=R/M$ as a linear space
$_\Delta\mathfrak M$ over the division ring $\Delta =\lambda(M,M)/M\cong\operatorname{End}(\mathfrak M)$. Then
$,$ for any nonempty finite linearly independent subset
$\{i_j\mid1\leqslant j\leqslant k\}$ of the linear subspace
$\lambda(M,P)/M=_\Delta\mathfrak B,$ and any other subset
$\{n_j\mid1\leqslant j\leqslant k\}$of
$_\Delta\mathfrak M,$ there is always an element
$r\in R$ such that
$1\leqslant j\leqslant k\ \Rightarrow\ n_j=i_jr$.}
It is not hard to see that for
$P=0$ theorem 1 turns into the well-known characterization of the Jacobson radical, and the density theorem turns into the usual Jacobson–Chevalley density theorem for primitive rings.
Bibliography: 4 titles.
UDC:
512.55
MSC: 16A20,
16A21 Received: 16.04.1987