Abstract
Let L be the Leavitt path algebra of an arbitrary directed graph E over a field K. This survey article describes how this highly non-commutative ring L shares a number of the characterizing properties of a Dedekind domain or a Prüfer domain expressed in terms of their ideal lattices. Special types of ideals such as the prime, the primary, the irreducible, and the radical ideals of L are described in terms of the graphical properties of E. The existence and the uniqueness of the factorization of a non-zero ideal of L as an irredundant product of prime or primary or irreducible ideals are established. Such factorization always exists for every ideal in L if the graph E is finite or if L is two-sided Artinian or two-sided Noetherian. In all these factorizations, the graded ideals of L seem to play an important role. Necessary and sufficient conditions are given under which L is a generalized ZPI ring, that is, when every ideal of L is a product of prime ideals. Intersections of various special types of ideals are investigated and an analogue of Krull’s theorem on the intersection of powers of an ideal in L is established.
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My thanks to Gene Abrams for carefully reading this article, making corrections, and offering suggestions.
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Rangaswamy, K.M. (2020). The Multiplicative Ideal Theory of Leavitt Path Algebras of Directed Graphs—A Survey. In: Facchini, A., Fontana, M., Geroldinger, A., Olberding, B. (eds) Advances in Rings, Modules and Factorizations. Rings and Factorizations 2018. Springer Proceedings in Mathematics & Statistics, vol 321. Springer, Cham. https://doi.org/10.1007/978-3-030-43416-8_16
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