Eugene Spiegel

Codes over Zm

In a previous paper (Codes over certain rings, Inform. Contr. 20, 396–404) Blake defined codes over Zm, the integers modulo m, if m is the product of distinct primes, and remarked that no codes were available over Zpn if n > 1, since Zpn is not a semisimple ring. We show that codes over Zpn can be described in terms of codes over Zp and thus are able to describe the codes over Zm for any positive m.

Codes over Zm, revisited

BCH codes are constructed over integer residue rings by using BCH oces over both p-adic finite fields.

Sums of projections

Necessary and sufficient conditions are given for an n × n complex matrix to be the sum of projection matrices. The result is applied to show when a matrix is the sum of involutions.

Minimal splitting fields in cyclotomic extensions

Suppose G is a finite group of exponent n and X an irreducible character of G. In this note we give sufficient conditions for the existence of a minimal degree splitting field L with Q(X) C L C Q(gn).

Isomorphisms ofp-adic group rings

SupposeP is the ring ofp-adic integers,G is a finite group of orderpn, andPG is the group ring ofG overP. IfVp(G) denotes the elements ofPG with coefficient sum one which are of order a power ofp, it is shown that the elements of any subgroupH ofVp(G) are linearly independent overP, and if in additionH is of orderpn, thenPG≅PH. As a consequence, the lattice of normal subgroups ofG and the abelianization of the normal subgroups ofG are determined byPG.

Upper-Triangular Embeddings of Incidence Algebras with Involution

Let X be a partially ordered set of order n, F a field, and I(X, F) the incidence algebra of X over F. If X has a 0 element and I(X, F) admits an involution, we determine when there is an involution preserving isomorphism of the incidence algebra into the n × n upper triangular F-matrices.

The similarity class of a matrix

If A is an n × n matrix over an infinite field F, k is a positive integer, and R is an arbitrary n × k matrix R1R2 over the field F, where R2 is a nonsingular k × k matrix, we give a necessary and sufficient condition which guarantees that the similarity class of A contains [R, S] for some n × (n − k) matrix S over the field F. This result extends a result of Barria and Halmos.

Group Gradings in Incidence Algebras

If X is a bounded countable locally finite partially ordered set, R an integral domain, and G a group having the property that no non-identity element has order a unit of R. Then it is shown that any G-grading of the incidence algebra I(X, R) is equivalent to a good grading. Further, an example is given showing that not all group gradings of incidence algebras are equivalent to good gradings.

Torsion Elements in Incidence Algebras

Abstract Let I(X, F) denote the incidence algebra of the locally finite partially ordered set, X, defined over the field, F. This paper considers when the set of torsion elements of I(X, F) forms a group. If F is of finite characteristic, it is shown that the torsion elements of I(X, F) form a group if and only if X is bounded, while if F is of characteristic 0, the torsion elements of the incidence algebra form a group if and only if X is an antichain.

PI INCIDENCE ALGEBRAS AND CAPELLI POLYNOMIALS

f, g ∈ I (X, R), r ∈ R, x, y, z ∈ X, and that X is locally finite if [x, y] = {z ∈ X | x ≤ z ≤ y} is finite for each x, y ∈ X . If x ≤ y ∈ X , then ex,y denotes that element of I (X, R) given by ex,y(x, y) = 1 and ex,y(u, v) = 0 if (x, y) = (u, v). Call x1, x2, . . . , xn ∈ X a chain of length n if x1 < x2 < · · · < xn . The poset X is of bound n if it has a chain of length of n and does not have any chain of length n + 1. If X is a chain of length n then it is a standard fact that I (X, R) is isomorphic to U Tn(R), the ring of all n × n upper triangular R-matrices (see, for example, [10]). Several authors (see, [4], [6], [7]) have shown that I (X, R) satisfies a polynomial identity of degree 2n, and no lower