Diagonability of idempotent matrices

On finitely generated projective ideals in commutative rings

“…An ordered pair (a, b) of elements of R is called semi-regular if there exists a 2 × 2 matrix with elements in R such that In fact M is idempotent (see [1]). In the last part of the paper we study finitely generated projective ideals in ID-rings and we show that such ideals are always principal thus strengthening Theorem 3 of [6]. In this work we generalize these results to ideals generated by ~ elements, and hence we give a new characterisation of finitely generated projective ideals in commutative rings with identity; see Theorem 1.2 (the abstract of this result appeared in [5]).…”

“…Since e v e r y quasi semi local r i n g is an I D ring [6], we get COROT,T,AR¥ 2.8. E v e r y f i n i t e l y generated projective ideal i n a q u a s i -s e m ilocal r i n g is p r i n c i p a l .…”

“…Since every invertible ideal is finitely generated and projective [4], we get the above mentioned result which appeared as Theorem 3 in [6], as a corollary:…”

“…Since I is projective, there exists an i d e m p o t e n t n × n m a t r i x M with elements in R such t h a t U M -= U a n d U ± -: a n n M. Since R is an I D ring, it follows f r o m [6] B y Proposition 1.5, I is g e n e r a t e d b y U P . L e t n be the smallest n u m b e r of generators t h a t are required to generate I, a n d let 1 be g e n e r a t e d b y U -(a 1, a~ .…”

“…It was proved in [6] that every invertible ideal I in an ID ring is principal. R is called an ID ring if every idempotent matrix with elements in R is equivalent to a diagonal matrix, i.e., if A is an n × n idempotent matrix with elements in R, there exists an invertible n × n matrix P with elements in R such that PAP-1 _--D, where D is a diagonal matrix.…”