Abstract This note draws conclusions that arise by combining two recent papers, Anuj Dawar, Erich Grädel and Wied Pakusa, published at ICALP 2019, Moritz Lichter, LICS 2021. In both the main technical results rely on combinatorial algebraic analysis of invertible-map equivalences ${\equiv ^{\text {IM}}_{k, Q}}$ certain variants Cai–Fürer–Immerman structures (CFI-structures for short). These Q}}...

All rings in this paper are commutative with unity; we will deal mainly with integral domains. Let R be a ring with total quotient ring K. A fractional ideal I of R is invertible if II−1 = R; equivalently, I is a projective module of rank 1 (see, e.g., [Eis95, Section 11.3]). Here, I−1 = (R : I) = {x ∈ K |xI ⊆ R}. Moreover, a projective R-module of rank 1 is isomorphic to an invertible ideal. (...

An n n sign pattern H is said to be sign-invertible if there exists a sign pattern H 1 (called the sign-inverse of H) such that, for all matrices A 2 Q(H), A 1 exists and A 1 2 Q(H 1). If, in addition, H 1 is sign-invertible (implying (H 1) 1 = H), H is said to be fully sign-invertible and (H;H 1) is called a sign-invertible pair. Given an n n sign pattern H, a Symplectic Pair in Q(H) is a pair...

primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. in fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly.  in this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pr...