### A structure theory for semialgebraic extensions of division algebras.

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A ring R is (in Vámos' terminology) if every subdirectly irreducible factor ring R/I is self-injective. rings include Noetherian rings, Morita rings and almost maximal valuation rings ([V1]). In [F3] we raised the question of whether a polynomial ring R[x] over a ring R is again . In this paper we show this is not the case.

Zelmanowitz [12] introduced the concept of ring, which we call ^{⊥} of a subset X of R is zero, then X_{1}
^{⊥} = 0 for a finite subset X_{1} ⊆ X.
(ZIP 2) If L is a left ideal and if L^{⊥} = 0, then L_{1}

We list some typos and minor correction that in no way affect the main results of (Publicacions Matemàtiques 33, 2 (1989), pp. 329-338), e.g., nothing stated in the abstract is affected.

In this paper we study a condition right FGTF on a ring R, namely when all finitely generated torsionless right R-modules embed in a free module. We show that for a von Neuman regular (VNR) ring R the condition is equivalent to every matrix ring R is a Baer ring; and this is right-left symmetric. Furthermore, for any Utumi VNR, this can be strengthened: R is FGTF iff R is self-injective.

A ring R is a ^{α} E = 0; equivalently,...

Addendum to the author's article "Rings whose modules have maximal submodules", which appeared in Publicacions Matemàtiques 39, 1 (1995), 201-214.

This paper owes its origins to Pere Menal and his work on Von Neumann Regular (= VNR) rings, especially his work listed in the bibliography on when the tensor product K = A ⊗ B of two algebras over a field k are right self-injective (= SI) or VNR. Pere showed that then A and B both enjoy the same property, SI or VNR, and furthermore that either A and B are algebraic algebras over k (see [M]). This is connected with a lemma in the proof of the , namely a finite ring extension K = k[a, ..., a] is...

A theorem of Utumi states that if R is a right self-injective ring such that every maximal ideal has nonzero annihilator, then R modulo the Jacobson radical J is a finite product of simple rings and is a von Neuman regular ring. We prove two theorems and a conjecture of Shamsuddin that characterize when R itself is a von Neumann ring, using a splitting theorem of the author on when the maximal regular ideal of a ring splits off.

In the article appeared in this same journal, vol. 33, 1 (1989) pp. 85-97, some statements in the proof of Example 3.4B got scrambled.

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