By Karlheinz Spindler

ISBN-10: 0824791444

ISBN-13: 9780824791445

A finished presentation of summary algebra and an in-depth therapy of the functions of algebraic suggestions and the connection of algebra to different disciplines, akin to quantity idea, combinatorics, geometry, topology, differential equations, and Markov chains.

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**Example text**

Then, since U is a uniform subsemigroup of U , there exists a ∈ U such that a Ha in Mn (D). If rank(as) = j, then we have a Mn (D) = as Mn (D) and Mn (D)a = Mn (D)a. 30 2. Prerequisites on semigroup theory Hence rank(a s) = j and asHa s. This means that the H-class of as in Mn (D) contains an element of S. The same holds for sa. A similar argument shows that, if z ∈ SG is of rank j and is in the H-class of an element of S, then, for any t ∈ S ∪ G, each of the elements tz, zt also is in the H-class of Mn (D) intersected by S, whenever it is of rank j.

6]). 6. Let G be a polycyclic-by-ﬁnite group and K a ﬁeld. The following properties hold. 1. clKdim(K[G]) = pl(G). 2. If K is absolute (that is, an algebraic extension of a ﬁnite ﬁeld) then every right primitive ideal M of K[G] is maximal and K[G]/M is ﬁnite dimensional. We ﬁnish with one more result ([103]). 7. A group algebra of a polycyclic-by-ﬁnite group is catenary. 4 Graded rings Let G be a group. A ring R is said to be G-graded if R = g∈G Rg , the direct sum of additive subgroups Rg of R, such that Rg Rh ⊆ Rgh for all 48 3.

1. A ring R graded by a polycyclic-by-ﬁnite group is right Noetherian if and only if R is graded right Noetherian. It is rather easy to verify that a minimal prime ideal of a Z-graded ring is homogeneous. In [74] this was extended to rings graded by a unique product group. Recall ([129]) that a group is said to be a unique product group if for any two nonempty subsets X and Y of G there exists a uniquely presented element of the form xy with x ∈ X and y ∈ Y . In [149] it is shown that such groups are two unique product groups, that is, given any two nonempty ﬁnite subsets X and Y of G with |X| + |Y | > 2 there exist at least two distinct elements g1 and g2 of G that have unique representations in the form g1 = x1 y1 , g2 = x2 y2 with x1 , x2 ∈ X and y1 , y2 ∈ Y .

### Abstract Algebra with Applications by Karlheinz Spindler

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