We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings it is proved that these are exactly the torsion T such that the regular module has a special T -preenvelope. In particular, every torsion enveloping class in Mod-R are of the form Gen(T ) for a minimal silting module T . For the dual case we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T ), where T is a cosilting module.

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We will study commuting properties of the defect functor Def β = CokerHom C (β, −) associate to a homomorphism β in a finitely presented category. As an application, we characterize objects M such that Ext 1 C (M, −) commutes with direct unions (i.e. direct limits of monomorphisms), assuming that C has a generator which is a direct sum of finitely presented projective objects.

A module M over a ring R satises the restricted minimum condition (RMC) if M=N is Artinian for every essential submodule N of M. A ring R satises right RMC if R R satises RMC as a right module. It is proved that (1) a right semiartinian ring R satises right RMC if and only if R=Soc(R) is Artinian, (2) if a semilocal ring R satises right RMC and Soc(R) = 0, then R is Noetherian if and only if the socle length of E(R=J(R)) is at most !, and (3) a commutative ring R satises RMC if and only if R=Soc(R) is Noetherian and every singular module is semiartinian.

The question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words. Recently, Aleksi Saarela has introduced a new approach to the problem that is based on linear-algebraic properties of polynomials encoding the equations and their solutions. In this paper we develop further this approach and take into account other algebraic properties of polynomials, namely their factorization. This, in particular, allows to improve the bound for the number of independent equations with maximal rank from quadratic to linear.Date: July 9, 2018. 1991 Mathematics Subject Classification. 68R15. Key words and phrases. independent systems of word equations, multivariate polynomials.