Gromov translation algebras over discrete trees are exchange rings

Abstract

Abstract. It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchange rings, including, for example, the algebras G(0) of ω × ω matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers r in the unit interval [0, 1], the growth algebras G(r) (introduced by Hannah and O'Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension r in [0,1].

Generating numbers of rings graded by amenable and supramenable groups

“…The ring $$T_G(G,R)$$ is denoted simply by $$T(G,R)$$. For a finitely generated group $$G$$, the rings $$T(G,R)$$ are the translation rings introduced by Gromov and investigated further in [6, 7, 15], and [34].…”

Generating numbers of rings graded by amenable and supramenable groups

“…The next phase of the proof involves translation rings of amenable groups with coefficients in a UGN-ring. Translation rings were introduced by M. Gromov [20] in 1993 for finitely generated groups and have attracted considerable attention (see, for example, [5], [6], [17], and [30]). For a ring S and a finitely generated group G, Gromov employed the right word metric d : G × G → N on G associated to some finite generating set in order to define the translation ring T (G, S) to be the ring of all G×G matrices M with entries in S for which there exists an integer r ≥ 0 such that M (x, y) = 0 whenever d(x, y) > r.…”

Characterizations of Gelfand rings specially clean rings and their dual rings

“…Clean rings have been extensively studied in the literature over the past and recent years, see e.g. [4], [5], [6], [7], [13], [17], [18], [24], [30], [35], [37], [39] and [43]. Of course the literature on clean rings is much more extensive than cited above.…”

“…Of course the literature on clean rings is much more extensive than cited above. But according to [5], although examples and constructions of exchange rings abound, there is a pressing need for new constructions to aid the development of the theory (note that in the commutative case, exchange rings and clean rings are the same things, see Theorem 5.3). Toward to realize this goal, then Theorem 5.3 can be considered as the culmination and strengthen of all of the results in the literature which are related to the characterization of commutative clean rings.…”