Ring M2 Research Articles

New singly and doubly even binary [72,36,12] self-dual codes from M2(R)G - group matrix rings

In this work, we present a number of generator matrices of the form [I2n|τ2(v)], where I2n is the 2n×2n identity matrix, v is an element in the group matrix ring M2(R)G and where R is a finite commutative Frobenius ring and G is a finite group of order 18. We employ these generator matrices and search for binary [72,36,12] self-dual codes directly over the finite field F2. As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings.

On a skew McCoy ring

A ring R with an endomorphism σ is called σ-skew McCoy, if for any zero-divisor f(x) in the skew polynomial ring R[x; σ], there exists a nonzero element with f(x)c = 0. In this note, we show that there exists a ring R and an endomorphism σ such that the matrix ring M2(R) is σ-skew McCoy. This gives a negative answer to the question posed in “A. R. Nasr-Isfahani, On semiprime right Goldie McCoy rings, Commun. Algebra 42 (2014) 1565-1570”.

Orbit Codes over M2(ℱq) and Their Homogeneous Distance

Let q be a power of a prime. The lattice of one-sided ideals of the finite unital non-commutative Frobenius ring M2(ℱq) of 2 × 2 matrices over the Galois field (ℱq) is completely analyzed. It turns out that M2(ℱq) is a principal left semi-local ring in which each left ideal is generated by an idempotent element. The explicit forms of the non-trivial idempotents of M2(ℱq) are determined to give q + 1 proper non-trivial left maximal ideals each with q elements. These are exactly the minimal left ideals as well. Using the structure of M2(ℱq) as a partial ordering of ideals, the generalized Möbius and Euler phi functions are applied to derive the explicit form of the homogeneous weight function on M2(ℱq). This weight depends on whether the element is the zero element, a zero divisor or a unit. A zero divisor gives the largest homogeneous weight. Moreover, orbit codes over M2(ℱq) are constructed via the action of the general linear group GL(2, q) on M2(ℱq) by left translation. The orbit determined by a nonzero nonunit idempotent element of M2(ℱq) forms the nonzero elements of a minimal left ideal of M2(ℱq) which are all zero divisors. Consequently, it is shown that the minimum homogeneous distance of the orbit code generated by a nonzero nonunit idempotent element of M2(ℱq) approaches the Plotkin upper bound as the field size q becomes larger. Analogous results are obtained when the lattice of right ideals is considered and the action of GL(2, q) on M2(ℱq) by right translation is used instead.

Bezout Rings of Stable Range 1.5

A ring R has a stable range 1.5 if, for every triple of left relatively prime nonzero elements a, b, and c in R, there exists r such that the elements a+br and c are left relatively prime. Let R be a commutative Bezout domain. We prove that the matrix ring M 2 (R) has the stable range 1.5 if and only if the ring R has the same stable range.

The Centralizer of an I-Matrix in M2(R/I), R a UFD

The concept of an I-matrix in the full 2 × 2 matrix ring M2(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an I-matrix [Formula: see text] in M2(R/I) as the sum of two subrings 𝒮1 and 𝒮2 of M2(R/I), where 𝒮1 is the image (under the natural epimorphism from M2(R) to M2(R/I)) of the centralizer in M2(R) of a pre-image of [Formula: see text], and the entries in 𝒮2 are intersections of certain annihilators of elements arising from the entries of [Formula: see text]. It turns out that if R is a PID, then every matrix in M2(R/I) is an I-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid.

A nil-clean 2 × 2 matrix over the integers which is not clean

While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ2(Z) which is not clean.

New Practical Algebraic Public-Key Cryptosystem and Some Related Algebraic and Computational Aspects

The most popular present-day public-key cryptosystems are RSA and ElGamal cryptosystems. Some practical algebraic generalization of the ElGamal cryptosystem is considered-basic modular matrix cryptosystem (BMMC) over the modular matrix ring M2(Zn). An example of computation for an artificially small number n is presented. Some possible attacks on the cryptosystem and mathematical problems, the solution of which are necessary for implementing these attacks, are studied. For a small number n, computational time for compromising some present-day public-key cryptosystems such as RSA, ElGamal, and Rabin, is compared with the corresponding time for the ВММС. Finally, some open mathematical and computational problems are formulated.

Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content

The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M2(F2) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M2(F2) in 90 geometric types. The classification over Klein’s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M2(F2), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version.

QUADRATIC FORMULAS FOR GENERALIZED QUATERNIONS

In this paper we derive explicit formulas for computing the roots of a quadratic polynomial with coefficients in a generalized quaternion algebra over any field 𝔽 with characteristic not 2. We also give some example of applications for the derived formulas, solving equations in the algebra of Hamilton's quaternions ℍ, in the ring M2(ℝ) of 2 × 2 square matrices over ℝ and in quaternion algebras over finite fields.

N-Clean Rings

Let n be a positive integer. A ring R is called n-clean if every element of R can be written as a sum of an idempotent and n units in R. The class of n-clean rings contains clean rings and (S,n)-rings (i.e., every element is a sum of no more than n units). In this paper, we investigate some properties on n-clean rings. There exists a clean and (S,3)-ring which is not an (S,2)-ring. If R is a ring satisfying (SI), then the polynomial ring R[x] is not n-clean for any positive integer n. An example shows that for any positive integer n> 1, there exists a non n-clean ring R such that the 2× 2 matrix ring M2(R) over R is n-clean.

The dimension monoid of a lattice

We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, $\Delta$ from L×L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Conc L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction: ¶¶ (1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to $\Bbb Z^+$ or to $\Bbb R^+$ .¶ (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power of ${\Bbb Z}^{+}\cup \{\infty\}.$ ¶ (3) If L is modular or if L has no infinite bounded chains, then Dim L is a refinement monoid.¶ (4) If L is a simple geometric lattice, then Dim L is isomorphic to $\Bbb Z^+$ , if L is modular, and to the two-element semilattice, otherwise.¶ (5) If L is an $\aleph_0$ -meet-continuous complemented modular lattice, then both Dim L and the dimension function $\Delta$ satisfy (countable) completeness properties.¶¶ If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M 2 (R), then Dim L is isomorphic to the monoid V (R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.

ON ESSENTIAL STRONG-NILPOTENCY OF γ-RINGS

An ideal I of a Γ-ring M is essential strongly-nilpotent if I contains a strongly-nilpotent ideal N of M such that K ∩ N ≠ 0 whenever K is a non-zero ideal of M contained in I. Let M be a Γ-ring in the sense of Nobusawa. The ring M2 = was defined by Kyuno. In this paper, the relationships between the unique largest essential strongly-nilpotent ideals of Γ-ring M and the corresponding ideals of the right operator ring R of M, the matrix T n,m -ring Mm,n , the M-ring T and the ring M2 are established.