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We introduce and study flipped non-associative polynomial rings. In particular, we show that all Cayley-Dickson algebras naturally appear as quotients of a certain type of such rings; this extends the classical construction of the complex numbers (and quaternions) as a quotient of a (skew) polynomial ring to the octonions, and beyond. We also extend some classical results on algebraic properties of Cayley-Dickson algebras by McCrimmon to a class of flipped non-associative polynomial rings., Comment: 17 pages; corrected an error; fixed some typos

Using the Rabinowitsch trick, we prove a version of Nullstellensatz over quaternions, which generalizes Hilbert's Nullstellensatz over complex numbers., Comment: 5 pages

The product formula for evaluating products of skew polynomials is used to construct a class of rings. As an application, we present a method of evaluating quotients of skew polynomials., Comment: arXiv admin note: text overlap with arXiv:2211.07006

We introduce a class of rings using which we define the concept of skew regularity for quaternion-valued functions over quaternions. It is shown that the notion of skew regularity coincides with the concept of slice regularity over symmetric slice domains. Known results regarding slice-regular functions over symmetric slice domains are generalized to skew-regular functions over general symmetric domains. Furthermore, we present new results concerning slice-regular functions.

The concept of a skew root of a skew polynomial is used to introduce notions of algebraic closedness for $\sigma$-fields, that is, a field equipped with an endomorphism. It is shown that every $\sigma$-field can be embedded in algebraically closed $\sigma$-fields of different types.

Rational twisted power series over a (commutative) field are studied. We give several characterizations of such series, which are similar to the classical results concerning rational power series over a commutative field. In particular, we prove a version of Kronecker's lemma for the rationality of twisted power series.

In this paper, the familiar notion of a Henselian pair is extended to the noncommutative case. Furthermore, the problem of Henselizations is studied in the noncommutative context, and it is shown that every (not necessarily commutative) pair which is Hausdorff with respect to a certain topology has a left (and right) Henselization.

In the first two papers, the author embarked on a study of classes of linear equations over integers satisfying a "Farkas-type" property. As the third paper in this study, the present paper deals with another class of linear equations over integers that has a similar "Farkas-type" property. Furthermore it is shown that if an arbitrary system of equations over integers satisfies the conditions imposed by Farkas' lemma then it has rational solutions of a special type.

In one of his papers, the author introduces the class of Farkas-related vectors for which a version of Farkas' lemma over integers is derived. In this paper, two similar classes are introduced and studied.

The main point of this paper is to present a class of equations over integers that one can check if they have a solution by checking a set of inequalities. The prototype of such equations is the equations appearing in the well-known Gale-Ryser theorem.

Latin squares are interesting combinatorial objects with many applications. When working with Latin squares, one is sometimes led to deal with partial Latin squares, a generalization of Latin squares. One of the problems regarding partial Latin square and with applications to Latin squares is whether a partial Latin square with a given set of conditions exists. The goal of this article is to introduce some problems of this kind and answer some existence questions regarding partial Latin squares, Comment: 5 pages

The pasch configuration and Steiner triple systems

A characterization of multiplicative (and additive) arithmetical functions is given. Using this characterization, we show that the group of multiplicative arithmetical functions is isomorphic to the group of additive arithmetical functions., Comment: 6 pages

By simulating an ergodic Markov chain whose stationary distribution is uniform over the space of nxn Latin squares, Mark T. Jacobson and Peter Matthews [4], have discussed elegant methods by which they generate Latin squares with a uniform distribution (approximately). The central issue is the construction of "moves" that connect the squares. Most of their lengthy paper is to prove that the associated graph is indeed connected. We give a short proof of this fact by using the concepts of Latin bitrades., Comment: 14 pages; The version to be published in Bull. Inst. Combin. Appl.

In this paper, the notion of F-schemes, a "generalization" of schemes, is introduced to cover unitary noncommutative rings., Comment: This is a more complete version than the previous one (with some changes in notations)

We study the self-dual Yang-Mills equations in split signature. We give a special solution, called the basic split instanton, and describe the ADHM construction in the split signature. Moreover a split version of t'Hooft ansatz is described.

A version of the Penrose transform is introduced in the split signature. It relates the cohomological data with supports on the open subsets of the complex 3-projective space and kernel of differential operators on the (real) Grassmannian of 2-planes in the Euclidean 4-space. As an example we derive a cohomological interpretation of the so-called X-ray transform. Furthermore, a cohomological realization of the so-called "minimal" representation of SL(4,R) is given. We also present the split Penrose transform in split instanton backgrounds., Comment: 23 pages

In this paper we study some properties of the field of rational pseudo-differential operators on a field and some other related rings. As an application we reconstruct the Kac co-cycle on the Lie algebra of differential operators on a circle., Comment: 13 pages

Non-commutative Henselian rings are defined and it is shown that a local ring which is complete and separated in the topology defined by its maximal ideal is Henselian provided that it is almost commutative., Comment: 7 pages; Added references, email and postal address