Monoids over which all strongly flat left acts are regular

Let Sbe a monoid. It is shown that all strongly flat left S-acts are regular if and only if all left S-acts having the property (E) are regular if and only if Sis a left PP monoid and satisfies (FP₂).This result answers a question in Kilp and Knauer [5].     

On monoids over which all strongly flat cyclic right acts are projective

A new characterization of monoids over which all strongly flat cyclic right acts are projective (projective generators, free) is given. This research has been supported by the Estonian Science Foundation, Grant No. 930.     

Rings over which all modules are semiregular

For a ring A, it is proved that all A-modules are semiregular if and only if A is an Artinian serial ring and J ²(A) = 0. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 185–194, 2007.     

On free,projective, and strongly flat acts

Archiv der Mathematik -     

Rings over which all modules are I 0-modules. II

All right R-modules are I ₀-modules if and only if either R is a right SV-ring or R/I ⁽²⁾ (R) is an Artinian serial ring such that the square of the Jacobson radical of R/I ⁽²⁾ (R) is equal to zero.     

Rings over which all torsion theories cogenerated by simple modules are jansian

Summary It is given a caracterization of (associative) rings (with unit element) over which all torsion theories cogenerated by simple modules are jansian. In the commutative case it is shown that such rings are finite products of local rings.

---

Riassunto Vengono caratterizzati gli anelli (associativi con unità) per i quali tutte le teorie di torsione cogenerate da moduli semplici sono Jansiane. Nel caso commutativo si prova che tali anelli sono prodotti finiti di anelli locali.

---

---

This work was written while the second author was visiting Università dell'Aquila, supported by a grant from the Consiglio Nazionale delle Ricerche.     

Not all quadrative norms are strongly stable

A norm N on an algebra A is called quadrative if N(x²) ≤ N(x)² for all x A, and strongly stable if N(x^k) ≤ N(x)^k for all x A and all k = 2, 3, 4…. Our main purpose in this note is to show that not all quadrative norms are strongly stable.     

On covers of cyclic acts over monoids

In (Bull. Lond. Math. Soc. 33:385–390, 2001) Bican, Bashir and Enochs finally solved a long standing conjecture in module theory that all modules over a unitary ring have a flat cover. The only substantial work on covers of acts over monoids seems to be that of Isbell (Semigroup Forum 2:95–118, 1971), Fountain (Proc. Edinb. Math. Soc. (2) 20:87–93, 1976) and Kilp (Semigroup Forum 53:225–229, 1996) who only consider projective covers. To our knowledge the situation for flat covers of acts has not been addressed and this paper is an attempt to initiate such a study. We consider almost exclusively covers of cyclic acts and restrict our attention to strongly flat and condition (P) covers. We give a necessary and sufficient condition for the existence of such covers and for a monoid to have the property that all its cyclic right acts have a strongly flat cover (resp. (P)-cover). We give numerous classes of monoids that satisfy these conditions and we also show that there are monoids that do not satisfy this condition in the strongly flat case. We give a new necessary and sufficient condition for a cyclic act to have a projective cover and provide a new proof of one of Isbell’s classic results concerning projective covers. We show also that condition (P) covers are not unique, unlike the situation for projective covers.     

关于所有强平坦右S-系是正则系的幺半群(英)

本论文考虑了所有强平坦右S-系是正则系的幺半群的刻画,证明了所有强平坦右S-系是正则S-系当且仅当S是右PSF幺半群并且S的每一个左coilpasible子幺半群包含左零元．该结果对Kilp和Knauer在文献[7]中的问题给出了一个新的回答．     

On Monoids for Which all Condition (P) Acts are Strongly Flat

In this article, we continue to investigate the monoids over which all right S-acts satisfying condition (P) are strongly flat, and we obtain some new classes of monoids, thereby extending all previous results in this area.     

Ordered semigroups which are both right commutative and right cancellative

In this paper we prove that each right commutative, right cancellative ordered semigroup (S,.,??) can be embedded into a right cancellative ordered semigroup (T,??,?) such that (T,??) is left simple and right commutative. As a consequence, an ordered semigroup S which is both right commutative and right cancellative is embedded into an ordered semigroup T which is union of pairwise disjoint abelian groups, indexed by a left zero subsemigroup of?T.