By Lopez-Permouth S., Huynh D.V. (eds.)

This quantity includes refereed learn and expository articles by means of either plenary and different audio system on the foreign convention on Algebra and functions held at Ohio collage in June 2008, to honor S.K. Jain on his seventieth birthday. The articles are on a large choice of parts in classical ring concept and module conception, resembling jewelry gratifying polynomial identities, jewelry of quotients, workforce earrings, homological algebra, injectivity and its generalizations, and so on. integrated also are functions of ring concept to difficulties in coding conception and in linear algebra.

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**Example text**

Proof. Let us assume that sext (R) = sext(R-simp). 5. As sext (R) is the class closed under extensions generated by the left ideals the hypothesis implies that each left ideal is semiartinian and ﬁnitely generated. Thus R is left noetherian and left semiartinian. Hence R is left artinian. If S is a simple module, then S ∈ sext (R) , thus S ∈ E(L(R), L(R))n for some minimal n ∈ N. So there exists an exact sequence 0 → I −→ S −→ K −→ 0 where I ∈ L(R) and K ∈ E(L(R), L(R))n−1 . As S is simple this implies that S ∼ = I.

On Big Lattices of Classes of R-modules 25 Suppose that M ∈ E(C, C)k for some 0 < k ∈ N, and let N be a submodule of M . Thus there exists an exact sequence M →0 0→L →M L M with L ∈ C, and ∈ E(C, C)k−1 . So we have the following commutative diagram: L M 0 → L → M → 0 L inc ↑ inc inc ↑ ↑ N +L → 0. 0 → L ∩ N → N L Since C and E(C, C)k−1 are hereditary classes, then N ∈ E(C, C)k . Thus for each n ∈ N, E(C, C)n is a hereditary class and it is immediate that n n∈N E(C, C) is also hereditary. Now we claim that n∈N E(C, C)n is closed under extensions.

Baer right ring hull of a semiprime ring with only ﬁnitely many minimal prime ideals. The existence of PP right ring hulls for reduced rings is also obtained. Further application to ring extensions such as monoid rings, matrix, and triangular matrix rings are investigated. Moreover, examples and counterexamples are provided. Mathematics Subject Classiﬁcation (2000). Primary 16N60; Secondary 16S20, 16P70. Keywords. -Baer rings, quasi-Baer rings. Throughout all rings are associative rings with unity.