By R. Keith Dennis

ISBN-10: 3540119663

ISBN-13: 9783540119661

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**Example text**

Aj ∈ K n . The following well known observations will be very useful, see [11]. 6. Λj : Mn (K) −→ M(n) (K) is a semigroup homomorphism j and the following assertions hold. 1. rank(Λj (a)) = r j if rank(a) = r ≥ j, and rank(Λj (a)) = 0 otherwise. 2. For every vectors a1 , . . , aj ∈ K n and every aik ∈ K, i, k = 1, . . , j, ( jk=1 a1k ak ) ∧ · · · ∧ ( jk=1 ajk ak ) = det(aik )(a1 ∧ · · · ∧ aj ). 3. If V ⊆ K n is a subspace with a basis f1 , . . , fj , and a ∈ Mn (K) is a matrix of rank j which determines an isomorphism a|V : V −→ V , then Λj (a)(f1 ∧ · · · ∧ fj ) = det(a|V )(f1 ∧ · · · ∧ fj ).

N. Moreover, Sj /Sj−1 is isomorphic to a completely 0simple semigroup M(Gj , nj , nj ; Ij ), where Gj is the group of all monomial matrices in GLj (D) and Ij is the identity matrix. 3. Let S ⊆ Mn (K) be a Zariski closed connected monoid over a ﬁeld K with group of units G. Then S is π-regular and the uniform components of S are of the form GaG, where rank(a) = rank(a2 ), see [133]. 4. Assume that Mn (D) is such that there exists an inﬁnite triangular set of idempotents T in some Mj / Mj−1 , j < n (division algebras D of this type were constructed in [139], see [123]).

4. Let R be ﬁnitely generated semiprime algebra over a ﬁeld K. If GK(R) = 1 then R is a ﬁnite module over its center. 6]). 5. Let K be a ﬁeld. If R is a K-algebra then GK(R[x1 , . . , xn ]) = GK(R) + n. Gromov [60] proved that the Gelfand-Kirillov dimension of a ﬁnitely generated group algebra K[G] is ﬁnite if and only if the group G is nilpotentby-ﬁnite. Earlier Bass proved a formula for this Gelfand-Kirillov dimension ([8]). Later Grigorchuk extended this to semigroup algebras of ﬁnitely generated cancellative semigroups [59].

### Algebraic K-theory. Proc. Oberwolfach 1980 by R. Keith Dennis

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