Vol. 55, No. 1, 1974

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Linear GCD equations

David Jacobson

Vol. 55 (1974), No. 1, 177–193

Let R be a GCD domain. Let A be an m×n matrix and B an m× 1 matrix with entries in R. Let c0,d R. We consider the linear GCD equation GCD(AX + B,c) = d. Let S denote its set of solutions. We prove necessary and sufficient conditions that S be nonempty. An element t in R is called a solution modulus if X + tRn S whenever X S. We show that if c∕d is a product of prime elements of R, then the ideal of solution moduli is a principal ideal of R and its generator t0 is determined. When R∕t0R is a finite ring, we derive an explicit formula for the number of distinct solutions ( mod t0) of GCD(AX + B,c) = d.

Mathematical Subject Classification 2000
Primary: 13G05
Secondary: 10B05
Received: 22 February 1974
Revised: 3 September 1974
Published: 1 November 1974
David Jacobson