Quasiregularity and primitivity relative to right ideals of a ring

V. A. Andrunakievich   ; Yu. M. Ryabukhin

Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 445-464

Résumé

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.

Bibliographie

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[2] Kherstein I., Nekommutativnye koltsa, Mir, M., 1972 | MR

[3] Andrunakievich V. A., Ryabukhin Yu. M., Radikaly algebr i strukturnaya teoriya, Nauka, M., 1979 | MR

[4] Andrunakievich V. A., Ryabukhin Yu. M., “Radikalnye zamykaniya pravykh idealov assotsiativnykh kolets”, Matem. issledovaniya, 38 (1976), 3–70 | Zbl