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On intersections of two-sided ideals of Leavitt path algebras
Rings and Algebras · Submitted 29 Oct 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It is shown that every ideal of $L$ is an intersection of primitive/prime ideals in $L$ if and only if the graph $E$ satisfies Condition (K).
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The abstract, sentence by sentence
Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. It is shown that every ideal of $L$ is an intersection of primitive/prime ideals in $L$ if and only if the graph $E$ satisfies Condition (K). Uniqueness theorems in representing an ideal of $L$ as an irredundant intersection and also as an irredundant product of finitely many prime ideals are established. Leavitt path algebras containing only finitely many prime ideals and those in which every ideal is prime are described. Powers of a single ideal $I$ are considered and it is shown that the intersection ${\displaystyle\bigcap\limits_{n=1}^{\infty}}I^{n}$ is the largest graded ideal of $L$ contained in $I$. This leads to an analogue of Krull's theorem for Leavitt path algebras.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. | background 0.99 | 0.10 | 0.02 | unclear 0.68 |
| s1 | It is shown that every ideal of $L$ is an intersection of primitive/prime ideals in $L$ if and only if the graph $E$ satisfies Condition (K). | result 1.00 | 0.91 | 0.70 | asserted 1.00 |
| s2 | Uniqueness theorems in representing an ideal of $L$ as an irredundant intersection and also as an irredundant product of finitely many prime ideals are established. | result 0.83 | 0.93 | 0.16 | asserted 1.00 |
| s3 | Leavitt path algebras containing only finitely many prime ideals and those in which every ideal is prime are described. | result 0.81 | 0.89 | 0.06 | asserted 0.99 |
| s4 | Powers of a single ideal $I$ are considered and it is shown that the intersection ${\displaystyle\bigcap\limits_{n=1}^{\infty}}I^{n}$ is the largest graded ideal of $L$ contained in $I$. | result 0.87 | 0.92 | 0.12 | asserted 1.00 |
| s5 | This leads to an analogue of Krull's theorem for Leavitt path algebras. | interpretation 0.92 | 0.79 | 0.10 | asserted 0.94 |
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Original abstract
Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. It is shown that every ideal of $L$ is an intersection of primitive/prime ideals in $L$ if and only if the graph $E$ satisfies Condition (K). Uniqueness theorems in representing an ideal of $L$ as an irredundant intersection and also as an irredundant product of finitely many prime ideals are established. Leavitt path algebras containing only finitely many prime ideals and those in which every ideal is prime are described. Powers of a single ideal $I$ are considered and it is shown that the intersection ${\displaystyle\bigcap\limits_{n=1}^{\infty}}I^{n}$ is the largest graded ideal of $L$ contained in $I$. This leads to an analogue of Krull's theorem for Leavitt path algebras.
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See for example the recent papers on the generators of ideals [ref] , prime and primitive ideals [ref] , two-sided chain conditions [ref] , and on irreducible representations [ref] .
Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras · Introduction · 0.96
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