Monoids characterized by their quasi-injective s-systems

• DOI: 10.1007/bf02575022
• Publication Date: 2007-03-22T17:34:05Z
• Authors: J. Ahsan
• Keywords: 510.mathematics; 0202 electrical engineering, electronic engineering, information engineering; 02 engineering and technology; Article

In this paper, S will denote a monoid, that is, a semigroup with identity i, which contains a zero 0. Each S-system is asumed to be right unitary (i.e. M1 = M) and centered (i.e. m0 = 0s = 0 for the zero 0 of M). A right S-system M S is injective if and only if, for every S-monomorphism g: Ps + Qs and for every Shomomorphism h: PS + MS' there exists an S-homomorphism h : QS + MS such that h g = h. Injective S-systems were originally introduced by Berthiaume [I], and later studied by many authors. Further references to this subject can be found in [4]. An S-system M S is quasi-injective if for N S ~ M S and every S-homomorphism f: N S + MS, there exists an S-homomorphism g: M S § M S such that giN = f" Every injective S-system is quasi-injective, but the converse is false. Quasiinjective S-systems have been studied by Lopez and Luedeman [6], and Satyanarayana [8]. Skornjakov ([10],Theorem 2) has proved that each direct sum of injective S-systems is injective if and only if S satisfies the ascending chain condition on right ideals. This result is not true if injectives are replaced by quasi-injectives. One object of this paper is to give a characterization of monoids whose class of quasi-injective S-systems is closed under the formation of direct sums. We shall also consider monoids all of whose S-systems are quasi-injectire. 2. PRELIMINARIES