## Download e-book for kindle: Group Representations and Special Functions (Mathematics and by A. Wawrzyńczyk

By A. Wawrzyńczyk

ISBN-10: 9027712697

ISBN-13: 9789027712691

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**Sample text**

Eld of R). This is, of course, the “usual” deﬁnition of the rank for ﬁnitely generated modules over such integral domains. We say that a commutative ring R is semilocal if R has only a ﬁnite number of maximal ideals. 8) (in the commutative case). 3. 7. Let R be a commutative semilocal ring. If P ∈ P(R) has constant rank n, then P ∼ = Rn . Proof. Suppose Max R = {m1 , . . , mr }. For any i (1 i r), pick xi1 , . . , xin ∈ P such that their images in Pmi form a free basis for Pmi . By the Chinese Remainder Theorem, there exist xj ∈ P (1 j n) such that xj ≡ xij (mod mi P ) for all 1 i r.

This proves (A). For (B), we ﬁrst handle the crucial case n = 2, following a calculation in [GillmanHenriksen: 1956]. 25)(5), we ﬁrst write (b1 , b2 ) = d · (x1 , x2 ), where d ∈ R and (x1 , x2 ) ∈ Um2 (R). From e = b1 R + b2 R = dR, we can write d = ke and e = d (for suitable k, ∈ R). Fixing an equation sx1 + tx2 = 1, let y1 = x1 k + rt and y2 = x2 k − rs, where r := k − 1. Since r kills d and e, we have yi e = xi ke = xi d = bi (for i = 1, 2). Finally, (s − x2 )y1 + (t + x1 )y2 = k (sx1 + tx2 ) − r(sx1 + tx2 ) = (k − r)(sx1 + tx2 ) = 1 shows that (y1 , y2 ) ∈ Um2 (R), as desired.

Of course (1) ⇒ (3). For (3) ⇒ (2), consider any β = (b1 , . . , bn ) with n > 2. Right multiplication by a suitable invertible matrix takes β to (b1 , . . , bn−2 , d, 0) for some d. Thus, (2) follows by induction on n. (2) ⇒ (1). Consider B ∈ IMm,n (R). There exists Q1 ∈ GLn (R) such that BQ1 has ﬁrst row (say) (d, 0, . . , 0). Let B1 be the matrix obtained from BQ1 by deleting the ﬁrst row and ﬁrst column. Invoking an inductive hypothesis (on m), we can right multiply B1 by a matrix in Q2 ∈ GLn−1 (R) to bring it to lower triangular form.

### Group Representations and Special Functions (Mathematics and its Applications) by A. Wawrzyńczyk

by Joseph

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