LAUSR

We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. We prove that if R is small, then the Weak Diamond implies that projectivity of an arbitrary module can be tested at the layer epimorphisms of R. The classic Baer’s Criterion, saying that a module M is injective, iff it is Rinjective, is a basic tool of the structure theory of injective modules over an arbitrary ring R. However, unless R is a perfect ring, there are no criteria available for the dual case, that is, for testing projectivity using a set of epimorphisms, [10]. For many non-perfect rings, it can be proved that there exist small (e.g., countably generated) non-projective R-projective modules (e.g., when R is commutative noetherian of Krull dimension ≥ 1). However, for each cardinal κ, there exists a non-right perfect ring Rκ such that all ≤ κ-generated Rκ-projective modules are projective, [12]. This is the best one can achieve in ZFC, because it is consistent with ZFC + GCH that if R is not right perfect, then there always exist (large) R-projective modules that are not projective, cf. [1]. Consistency (and hence independence) of the coincidence of R-projectivity and projectivity for certain commutative non-noetherian rings was proved in [11]. This answered in the positive a question from [1, 2.8], and clarified the set-theoretic status of an old problem by Carl Faith [6, p.175]. The consistency result was extended in [12] to further classes of rings that are finite Loewy length extensions of simple artinian rings. The set theoretic tool used in [11] and [12] was Jensen’s Diamond. The goal of the present paper is twofold: to enhance the algebraic tools to cover infinite Loewy length extensions of simple artinian rings, and to weaken the settheoretic assumptions used in the proofs. In Theorem 3.2 below, we show that the Weak Diamond Principle Φ and CH are sufficient to prove coincidence of the classes of all weak R-projective, R-projective, and projective modules, in the case when R has cardinality at most א1, its Loewy length is countable, and each proper layer of R is countably generated. That is, in this case, Φ and CH imply that the projectivity of a module M is equivalent to the factorization of all morphisms from M with finitely generated images through the layer epimorphisms of R. The latter are just the canonical projections πα : Sα+1 → Sα+1/Sα (α < σ), where (Sα | α ≤ σ + 1) is the socle sequence of R. For basic notions and facts needed from ring and module theory, we refer to [2] and [8]; our references for set-theoretic homological algebra are [4] and [7]. Date: March 14, 2022. 2010 Mathematics Subject Classification. Primary: 16D40, 03E35. Secondary: 16E50, 16D60, 16D70, 03E45, 18G05.