Ideals In Matrix Rings Research Articles

Left ideals of matrix rings and error-correcting codes

It is well-known that each left ideal in a matrix rings over a finite field is generated by an idempotent matrix. In this work we compute the number of left ideals in these rings, the number of different idempotents generating each left ideal, and give explicitly a set of idempotent generators of all left ideals of a given rank. We then apply these results to give examples of left group codes that have best possible minimum weight.

Matrix Rings over Reflexive Rings

The concept of reflexive property is introduced by Mason. This note concerns a ring-theoretic property of matrix rings over reflexive rings. We introduce the concept of weakly reflexive rings as a generalization of reflexive rings. From any ring, we can construct weakly reflexive rings but not reflexive, using its lower nilradical. We study various useful properties of such rings in relation with ideals in matrix rings, showing that this new property is Morita invariant. We also investigate the weakly reflexive property of several sorts of ring extensions which have roles in ring theory.

Left ideals and 0-primitivity in matrix near-rings

Maximal left ideals in matrix rings were studied by Stone [10]. Similar results are not necessarily valid in the general near-ring case and one of the objectives of this paper is to study these differences. Furthermore, although much is known about 2-primitivity in general matrix near-rings (Van der Walt [11]), quite the opposite is true for 0-primitivity and the other objective of this paper is to present some results on 0-primitivity in matrix near-rings in certain restricted cases.

Maximal Left Ideals and Idealizers in Matrix Rings

Let R be a ring with identity, Mn(R) the ring of n × n matrices over R. The lattice of two-sided ideals of R is carried via A → Mn(A) to form the lattice of two-sided ideals of Mn(R). We wish to study the more complex left ideal structure of Mn(R). For example, if K is a commutative field, then Mn(K) has non-trivial left ideals. In particular Mn(K) has the maximal left ideal consisting of all matrices with some designated column zero. Or for any ring with maximal left ideal M, Mn(R) has the maximal left ideal consisting of all matrices with some column's entries from M. In Theorem 1.2 we characterize the maximal left ideals of Mn(R) in terms of those of R. We briefly study some contraction properties of maximal left ideals in matrix rings. For R commutative we “count” the maximal left ideals of Mn(R) and describe the idealizer of any such ideal; in the case where K is a field we see that the collection of maximal left ideals of Mn(K) can be naturally identified with Pn–1(K) (projective space).

Ideals of matrixrings over nonassociative rings

Ideals of matrixrings over nonassociative rings

Solvable normal subgroups and nilpotent ideals in matrix rings

Solvable normal subgroups and nilpotent ideals in matrix rings

Principal ideals in matrix rings

Let be a ring with a unity 1, and let n be a positive integer. It is well-known [3, p. 37]1 that every two-sided ideal of R (the complete matrix ring of order n over R) is necessarily of the form M, where M is a two-sided ideal of R. Simple examples show that this result no longer holds for one-sided ideals. In this note we investigate the left ideals of R in the case when is a principal ideal ring (an integral domain in which every ideal is principal). We shall prove THEOREM 1: IfR is a principal ideal ring, then every left ,ideal ofRn is principal_ The proof of Theorem 1 depends upon the fact that if A is any p X q matrix over R, then a unit matrix V of Rp exists such that the p X q matrix VA is upper triangular [2 , p. 32]_ We also establish the following partial converse to Theorem 1: THEOREM 2: If is not Noetherian or if is a Dedekind ring but not a principal ideal ring, then Rn contains a nonprincipalleft ideal_ For general information on rings , see [3]. For information on Dedekind rings, see [1 , p. 101].

Prime Ideals in Matrix Rings

Let R be a ring and let Rn be the complete ring of n × n matrices with coefficients from R.