SIFAT PRIMA DAN SEMIPRIMA MENDASAR ALJABAR LINTASAN DAN ALJABAR LINTASAN LEAVITT ATAS RING KOMUTATIF UNITAL PADA GRAF BERHINGGA ( ON BASICALLY PRIME AND SEMIPRIME PATH ALGEBRAS AND LEAVITT PATH ALGEBRAS OVER A UNITAL COMMUTATIVE RING ON A FINITE GRAPH )
Wardati, Khurul, Sri Wahyuni
2015 | Tesis | FMIPAThe initial idea of the dissertation came from the results of research on the necessary and sufficient conditions of a graph such that a Leavitt path algebra over a unital commutative ring (LR(E)) is basically simple. The definition of basically simple is based on the term of basic ideal in LR(E) that is a generalization of Leavitt path algebras over a field (LK(E)). The idea of developing the properties of LR(E) is based on the findings of the necessary and sufficient condition of a graph so that LK(E) is prime, and any LK(E) is semiprime. Similarly, since LK(E) is an extension of path algebras over a field (KE) then we need to discuss path algebras over a unital commutative ring (RE) as a generalization of KE: A discovery of the necessary and sufficient conditions on a graph, such that KE is a prime algebra, inspired the idea to examine the properties of RE: We need to discuss the free algebras because both path algebras and Leavitt path algebras are free algebras. The definition of basic ideal in free algebras is a generalization of the basic ideal in LR(E): An ideal in free algebras is called a free ideal if it has a basis contained in a basis of its algebra. The free ideal is a necessary and sufficient condition of a basic ideal. Both have an important role in developing the properties of the free algebras. Based on the basic ideal, we can define a (semi) prime basic ideal and the term of basically (semi) prime algebra. The main results of this study are the theorems of the necessary and sufficient conditions of a basic ideal that is a (semi) prime ideal, and the necessary and sufficient conditions of a free algebra that is basically (semi) prime. These theorems contribute in the proof of basically (semi) prime path algebras and Leavitt path algebras. There are some similarities between the properties of path algebras and Leavitt path algebras on a graph, both over a field and a unital commutative ring. Each of these is an associative algebra and a graded algebra. In addition, it is a unital algebra if the graph is finite, a finite-dimensional algebra if the finite graph is acyclic. These algebras on a finite graph have a same unity that is a sum of all vertices, but if the graph is infinite then the algebras have a local unit. An arrow ideal in RE on a connected finite graph is a graded basic ideal of all linear combinations of paths that are not vertices, then it does not contain any vertex. We can generalize the arrow ideal to an ideal I(X) in RE on a finite graph E that should not be connected, in which X ? E0 is a nonempty hereditary subset which has no isolated vertex. The graded basic ideal I(X) consists of all linear combinations of paths whose range in X and the paths are not vertices. The ideal I(X) contains basic ideals I(X)n2N that is an ideal of all linear combinations of paths whose range in X of length greater than or equal to n; so that I(X)n does not contain X for every n: Furthermore, we can define a graded basic ideal constructed by a hereditary subset H ? E0; namely IH that is an ideal of all linear combinations of not only paths whose range in H but also vertices in H: The graded basic ideal IH is a generalization of the graded basic ideal in LR(E) defined as an ideal of linear combinations of monomials ??? with ?; ? 2 Path(E) and r(?) = r(?) 2 H: If H ? E0 is saturated hereditary then a graded basic ideal IH in RE is prime if only ifM = E0nH is a maximal tail and for every path ? whose r(?) 2 M; there is a path ? such that r(?) = s(?) and s(?) = r(?): The second condition is a necessary and sufficient condition for semiprime basic ideal IH in RE: In addition, path algebra RE is basically prime if only if E0 is a maximal tail and for every path ?; there is a path ? such that r(?) = s(?) and s(?) = r(?): The second condition is a necessary and sufficient condition for basically semiprime RE: This is because I; is a zero basic ideal and ; is a saturated hereditary subset. If H ? E0 is saturated hereditary then a graded basic ideal IH in LR(E) is prime if only if M = E0nH is a maximal tail. A zero basic ideal in LR(E) is also equal to I; and E0 always satisfies the conditions MT1;MT2 then Leavitt path algebra LR(E) is basically prime if only if E0 is a maximal tail. It means that E0 satisfies the condition MT3; that is for every v;w 2 E0 there exists y 2 E0 such that v ? y and w ? y: In addition, any graded basic ideal in LR(E) is semiprime so that every Leavitt path algebra LR(E) is basically semiprime.
Kata Kunci : path algebra; Leavitt path algebra; basic ideal; free ideal; (semi) prime basic ideal; basically (semi) prime algebra; hereditary, saturated; maximal tail.
