Pairs of domains where all intermediate domains satisfy S-ACCP

The rings considered in this paper are commutative with identity. If [Formula: see text] is a subring of a ring [Formula: see text], then we assume that [Formula: see text] contains the identity element of [Formula: see text]. Let [Formula: see text] be a multiplicatively closed subset (m.c. subset) of a ring [Formula: see text]. An increasing sequence of ideals [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-stationary if there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]. This paper is motivated by the research work [A. Hamed and H. Kim, On integral domains in which every ascending chain on principal ideals is [Formula: see text]-stationary, Bull. Korean Math. Soc. 57(5) (2020) 1215–1229]. Let [Formula: see text] be a m.c. subset of an integral domain [Formula: see text]. We say that [Formula: see text] satisfies [Formula: see text]-ACCP if every increasing sequence of principal ideals of [Formula: see text] is [Formula: see text]-stationary. Let [Formula: see text] be a subring of an integral domain [Formula: see text] and let [Formula: see text] be a m.c. subset of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-ACCP pair if [Formula: see text] satisfies [Formula: see text]-ACCP for every subring [Formula: see text] of [Formula: see text] with [Formula: see text]. The aim of this paper is to provide some pairs of domains [Formula: see text] such that [Formula: see text] is an [Formula: see text]-ACCP pair, where [Formula: see text] is a m.c. subset of [Formula: see text].

Truncations of ordered abelian groups

We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a $$+$$ + , but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments $$[0, \tau ]$$ [ 0 , τ ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group $${\mathbb {Z}}$$ Z or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

On the number of principal ideals in d-tonal partition monoids

For a positive integer d, a non-negative integer n and a non-negative integer $$h\le n$$ h ≤ n , we study the number $$C_{n}^{(d)}$$ C n ( d ) of principal ideals; and the number $$C_{n,h}^{(d)}$$ C n , h ( d ) of principal ideals generated by an element of rank h, in the d-tonal partition monoid on n elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions.

Arithmetic of matrices over rings

The book is devoted to investigation of arithmetic of the matrix rings over certain classes of commutative finitely generated principal ideals do- mains. We mainly concentrate on constructing of the matrix factorization theory. We reveal a close relationship between the matrix factorization and specific properties of subgroups of the complete linear group and the special normal form of matrices with respect to unilateral equivalence. The properties of matrices over rings of stable range 1.5 are thoroughly studied. The book is intended for experts in the ring theory and linear algebra, senior and post-graduate students.

Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves

Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve E defined over 𝔽 p 2 , if an imaginary quadratic order O can be embedded in End(E) and a prime L splits into two principal ideals in O, we construct loops or cycles in the supersingular L-isogeny graph at the vertices which are next to j(E) in the supersingular ℓ-isogeny graph where ℓ is a prime different from L. Next, we discuss the lengths of these cycles especially for j(E) = 1728 and 0. Finally, we also determine an upper bound on primes p for which there are unexpected 2-cycles if ℓ doesn’t split in O.

On the dimension of ideals in group algebras, and group codes

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.