Serial Rings

Some model theory of modules over Bézout domains. The width

“…The implication holds true also over B P which is a valuation domain. It follows from [9,Cor. 12.4] that for some i we have both…”

“…Note that the condition (i) has already occurred for valuation domains (see [9,Prop. 12.11]), however (ii) and ( The condition (iii) is analyzed similarly.…”

“…For instance, the question on decidability of T (V ) was answered almost completely (see [12], [3], [5]) and the sobriety of Zg V (see [4]) was established. Furthermore (see [9,Ch. 12 In this paper we will extend this natural approach to commutative Bézout domains B.…”

Superdecomposable pure-injective modules

“…• Over serial rings, the model theory of modules and the structure of pure-injectives (as well as pure-projectives, that is, direct summands of direct sums of finitely presented modules) has been extensively studied by Puninski in a series of papers, see especially [45], [46] and the book [44]. If R is a serial ring Puninski gives a criterion, purely in terms of the structure of the lattices of right and left ideals of R, equivalent to existence of a superdecomposable pure-injective ( [50, 5.2]).…”

Classifying Projective Modules Over Some Semilocal Rings

“…Here a uniserial domain is said to be nearly simple, if J(R) is the only nonzero two-sided ideal of R. From point of view of finitely presented modules nearly simple uniserial domains are very alike to rings of finite representation type. Indeed (see [18,Ch. 14]) every finitely presented R-module is a direct sum of copies of R and R/rR, 0 = r ∈ J(R), and all the modules R/rR are isomorphic.…”