Jespers E., Okninski J.'s Noetherian Semigroup Algebras PDF
By Jespers E., Okninski J.
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Extra resources for Noetherian Semigroup Algebras
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. Therefore, an induction on the length of the elements of SG as words in S ∪ G allows to show that S and SG intersect the same H-classes of Mn (D) consisting of matrices of rank j.
5]). 2. Let G be a ﬁnitely generated group. If G is nilpotent or a ﬁnite conjugacy group then G is polycyclic-by-ﬁnite. In the latter case [G : Z(G)] < ∞. 46 3. Prerequisites on ring theory Next we state some well known facts on a polycyclic-by-ﬁnite groups. 4 in . 3. The following properties hold for a ﬁnitely generated group G. 1. If G is nilpotent, then the torsion part G+ of G is a ﬁnite normal subgroup. 2. Assume G is torsion free and nilpotent. Let x, y ∈ G. If, for some positive integer r, the elements xr and y r commute then x and y commute; and if xr = y r then x = y.
It turns out that the center Z(R) of a Krull order R is a Krull domain. The divisor group D(R) of a Krull order R is a free abelian group with basis the set of prime divisorial ideals. The latter are primes of height one and for rings satisfying a polynomial identity the height one primes are precisely the prime divisorial ideals. 6]). 1. Let R be a prime algebra satisfying a polynomial identity. Then, R is a Krull order if and only if R is a maximal order and Z(R) is a Krull domain. Moreover, in this case the following properties hold.
Noetherian Semigroup Algebras by Jespers E., Okninski J.