Matrix theory over ringoids
Abstract
In this paper, we study the notion of right distributive ringoids over a field which are neither rings, semi-rings nor near-rings. Matrices over ringoids are defined, and new concepts such as top-row-determinate and down-row-determinate related to a 2 × 2 matrices over a ringoid are
introduced. Moreover, we discuss on the notions of the (strongly, (very-) weak) orthogonality of vectors over a ringoid. Beside, we discuss on the notion of incident of vectors and define the concept of α-K-sphere on a ringoid, where K is a field and investigate some of their properties.
Finally, we show that in a commutative ringoid all of vectors are strongly orthogonal.
introduced. Moreover, we discuss on the notions of the (strongly, (very-) weak) orthogonality of vectors over a ringoid. Beside, we discuss on the notion of incident of vectors and define the concept of α-K-sphere on a ringoid, where K is a field and investigate some of their properties.
Finally, we show that in a commutative ringoid all of vectors are strongly orthogonal.
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