Group Gradings on Full Matrix Rings

Graded Rings and Graded Grothendieck Groups

“…By definition, this is a grading on M n (A) such that the matrix unit e i j , the matrix with 1 in the i j-position and zero everywhere else, is homogeneous, for 1 ≤ i, j ≤ n. This particular group gradings on matrix rings have been studied by Dȃscȃlescu et al [36] (see Remark 1.3.9). Therefore for x ∈ A,…”

“…More concretely, P is graded projective if for any diagram of graded modules and graded A-module homomorphisms 36) there is a graded A-module homomorphism h : P → M with gh = j. In Proposition 1.2.15 we give some equivalent characterisations of graded projective modules, including the one that shows an A-module is graded projective if and only if it is graded and projective as an A-module.…”

“…One can put a Γ-grading on the ring M n (K), by assigning a degree (an element of the group Γ) to each matrix unit e i j , 1 ≤ i, j ≤ n. This is called a good grading or an elementary grading. This grading has been studied in [36]. In particular it has been shown that a grading on M n (K) is good if and only if it can be described as M n (K)(δ 1 , .…”

Degree-inverting involutions on matrix algebras

“…Initially with a paper of Dăscălescu, et. al [7] and later with other authors and papers, e.g., [1,3,5,6].…”

Good Gradings of Generalized Incidence Rings

“…An important type of matrix algebra grading is a good grading (for example, see references [2], [3], and [4]). This definition extends easily to incidence algebras (see [4], [6], and [7]).…”