##
**\(\omega_1\)-generated uniserial modules over chain rings.**
*(English)*
Zbl 1101.16014

Recall, that an associative ring with identity is called a chain ring if the lattices of all left and right ideals are linearly ordered. A nearly simple ring is a ring having exactly three two-sided ideals, namely \(0\), \(R\) and the Jacobson radical \(J(R)\).

Over any non-Artinian nearly simple chain ring there exists an \(\omega_1\)-generated uniserial module (Theorem 1.8).

A module \(M\) is said to be dually slender if the functor \(\operatorname{Hom}_R(M,-)\) commutes with direct limits and the ring \(R\) is called right steady if the class of all dually slender modules coincides with the class of all finitely generated modules.

A chain ring \(R\) is right steady if and only if there exists no \(\omega_1\)-generated uniserial module and these conditions are equivalent to the fact that \(R/\text{rad}(R)\) (\(\text{rad}(R)\) denotes the prime radical of \(R\)) contains no uncountable strictly decreasing chain of ideals, \(R\) contains no uncountably generated right ideal and for every right ideal \(I\) and for every prime ideal \(P\subseteq I\) there exists an ideal \(K\) such that \(P\subset K\subset I\) (Theorem 2.4).

A ring \(R\) is called serial if it contains a set of orthogonal idempotents \(\{e_i,\;i\leq n\}\) such that \(1=\sum_{i\leq n}e_i\) and for every index \(i\leq n\) the ideals \(e_iR\) and \(Re_i\) are uniserial modules.

In the last part of the paper it is shown that the following conditions for a serial ring \(R\) with a complete set of orthogonal idempotents \(\{e_i,\;i\leq n\}\) are equivalent: (1) \(R\) is right steady; (2) \(e_iRe_i\) is right steady for every \(i\leq n\); (3) there is no \(\omega_1\)-generated uniserial right \(R\)-module (Theorem 3.5).

Over any non-Artinian nearly simple chain ring there exists an \(\omega_1\)-generated uniserial module (Theorem 1.8).

A module \(M\) is said to be dually slender if the functor \(\operatorname{Hom}_R(M,-)\) commutes with direct limits and the ring \(R\) is called right steady if the class of all dually slender modules coincides with the class of all finitely generated modules.

A chain ring \(R\) is right steady if and only if there exists no \(\omega_1\)-generated uniserial module and these conditions are equivalent to the fact that \(R/\text{rad}(R)\) (\(\text{rad}(R)\) denotes the prime radical of \(R\)) contains no uncountable strictly decreasing chain of ideals, \(R\) contains no uncountably generated right ideal and for every right ideal \(I\) and for every prime ideal \(P\subseteq I\) there exists an ideal \(K\) such that \(P\subset K\subset I\) (Theorem 2.4).

A ring \(R\) is called serial if it contains a set of orthogonal idempotents \(\{e_i,\;i\leq n\}\) such that \(1=\sum_{i\leq n}e_i\) and for every index \(i\leq n\) the ideals \(e_iR\) and \(Re_i\) are uniserial modules.

In the last part of the paper it is shown that the following conditions for a serial ring \(R\) with a complete set of orthogonal idempotents \(\{e_i,\;i\leq n\}\) are equivalent: (1) \(R\) is right steady; (2) \(e_iRe_i\) is right steady for every \(i\leq n\); (3) there is no \(\omega_1\)-generated uniserial right \(R\)-module (Theorem 3.5).

Reviewer: Ladislav Bican (Praha)

### MSC:

16L30 | Noncommutative local and semilocal rings, perfect rings |

16D25 | Ideals in associative algebras |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

16D80 | Other classes of modules and ideals in associative algebras |

16P70 | Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras) |