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ON PRIMENESS OF PATH ALGEBRAS OVER A UNITAL COMMUTATIVE RING
In this paper, we first discuss the primeness of basic ideals in a free R-algebra where R is a unital commutative ring. The condition of primeness is applied to show a prime basic ideal in a path algebra RE on a graph E. For every hereditary subset H, we can construct a (graded) basic ideal IH in RE. The basic ideal IH is an ideal of linear combinations of vertices in H and paths whose ranges in H. The main purpose of this paper is to present the necessary and sufficient conditions on a graph, so that IH is a prime basic ideal, if H is saturated hereditary. Since ∅ is saturated hereditary, we find the necessary and sufficient conditions on a graph, so that a path algebra RE is basically prime.
JP Journal of Algebra, Number Theory and Applications
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