The commutative ring of a symmetric design

✍ Scribed by Alan R. Prince

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 162 KB
• Volume: 80
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(90)90300-7

✦ Synopsis

It is shown that any symmetric design has associated to it a certain commutative ring , the author defined a commutative ring associated with any finite projective plane. The purpose of this paper is to define an analogous ring associated with any symmetric design. A symmetric (v, k, A)-design is an incidence structure consisting of u points and v blocks each of size k with the property that any two distinct points are in precisely il blocks and any two distinct blocks have precisely ;1 points in common (see [l] for further details).

Let X denote the set of points of a symmetric (v, k, )L)-design. The order n of the design is defined to be k -A. Adjoin a formal symbol 1 to X to yield a set X*. Let R denote the free Z-module whose basis is formed by the elements of X*. If B is a block, we denote the element CbeB b of R simply by B (the context should serve to distinguish between the two usages of the symbol). We obtain a multiplication on R by defining the product of the basis elements by the following rules, and then extending bilinearly:

(i) if a and b are distinct points of X, then ab=B,+B,+...+B,-nA21