Graded Rings and Graded Grothendieck Groups

Abstract

This study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra which has recently been adopted to classify the Leavitt path algebras. The book begins with a concise introduction to the theory of graded rings and then focuses in more detail on Grothendieck groups, Morita theory, Picard groups and K-theory. The author extends known results in the ungraded case to the graded setting and gathers together important results which are currently scattered throughout the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.

Baer and Baer *-ring characterizations of Leavitt path algebras

“…Conversely, any finitely generated graded free right A-module is of this form. A graded projective right A-module is a graded module which is graded isomorphic to a direct summand of a graded free right A-module (see [18,Proposition 1.2.15] for an equivalent definition).…”

“…A Leavitt path algebra can be graded by an arbitrary abelian group also (see [18] or [19]) and all of our results can easily be adapted to any other grading of a Leavitt path algebra.…”

“…(1) are locally Baer since the finite sums of matrix units e ii , i ∈ κ, constitute sets of local units such that the corners are isomorphic to M n (K) and to (2) are graded regular. One could check this directly or use the representations of these algebras as Leavitt path algebras over certain acyclic or comet graphs (see [18]) and then use [17,Theorem 9]. If K is positive definite, then these algebras are positive definite as well.…”

“…Namely, by choosing suitable graphs, one can produce prime rings which are not primitive (Kaplansky's Conjecture [4]), simple rings which are not purely infinite simple ( [2]), and strongly graded rings which are not crossed-products ( [18]), to mention just a few applications. The characterization theorems have been formulated and proven for a number of ring-theoretic properties: being simple, purely infinite simple, hereditary, exchange, semisimple, artinian, noetherian, directly finite, to name some of them, and, in particular, von Neumann regular ( [5]), right semihereditary and right nonsingular ( [8,10,23]).…”

Graded Chain Conditions and Leavitt Path Algebras of No-Exit Graphs

“…For the converse, if Ru i is graded noetherian for every i ∈ I, then any finitely generated graded free left module is a submodule of a direct sum of finitely many modules Ru i and, hence, graded noetherian. As a consequence, any graded left module with finitely many homogeneous generators (which is a graded homomorphic image of a finitely generated graded free left module, see [12,Section 1.2.4]) is graded noetherian too. If R is graded, then every noetherian module over R is also graded noetherian.…”