By Toma Albu, Robert Wisbauer (auth.), S. K. Jain, S. Tariq Rizvi (eds.)

ISBN-10: 1461219787

ISBN-13: 9781461219781

ISBN-10: 1461273641

ISBN-13: 9781461273646

"[The e-book] indicates advancements in lots of topics of this very lively box of jewelry and modules and it encompasses a wealth of latest rules, innovations and effects added by means of the most very important researchers within the field..."

**--Mathematica**

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PARK generates Mod-R. For examples and basic results for right GFC rings see [2] and for right FPF rings see [6]. J(R), peR), Soc(RR) , SOC(RR) , Z(RR), and Z(RR) denote the Jacobson radical, prime radical, left socle, right socle, left singular ideal, and right singular ideal of R, respectively. An ideal I of R is called ideal essential in an ideal H of R if I has nonzero intersection with every nonzero ideal of R contained in H. A right (resp. left) ideal X of R is called right (resp. left) essential in a right (resp.

Examples 8. Let A be a complete abelian category. COMPACTNESS IN CATEGORIES AND INTERPRETATIONS 25 i) Choosing F to be the identical functor of A, F-compactness means discrete linear compactness in the sense of [2] (an object X E Ob A is discrete linearly compact, if for every filter {Xo,} consisting of subobjects of X the canonical morphism X -------+ lim{X/Xo,} is an epimorphism). +-- ii) Starting from the category A we construct a category C by topologizing the objects of A. A topology on an object X E Ob A is a filter {X,:r} of subobjects X o ' The filter {Xo } induces a canonical morphism T: X -------+ limX/Xo and T determines uniquely the filter {X o }.

In the case of unary algebras (that is, S-acts) Normak [9], [10) calls this linear compactness as congruence compactness. 30 P. N. ANH AND R. WIEGANDT The authors are indebted to the referee for pointing out some shortcomings. REFERENCES 1. Anh, P. , An Embedding Theorem for Abelian Categories, J. Algebra 167 (1994), 627-633. 2. Anh, P. , Wiegandt, R, Morita Duality for Grothendieck Categories, J. Algebra 168 (1994), 273-293. 3. , Linearly Compact Algebras, Annales Vniv. Sci. Budapest. 21 (1978), 129-137.

### Advances in Ring Theory by Toma Albu, Robert Wisbauer (auth.), S. K. Jain, S. Tariq Rizvi (eds.)

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