Unimodular Elements and Projective Modules

We are mainly concerned with the completablity of unimodular rows to an invertible matrix. We first relate this to stable range of the ring. Further, we consider rows generating an idempotent of the ring to obtain completability under special conditions. We try to obtain analogues of important theorems regarding unimodular rows in the case of these “idempotent rows”. By ring, we shall always mean a commutative ring with identity. Section 1: Stable Range and Unimodular Elements We consider a homomorphism from R to R. The mapping may be represented as a row of length n. If we assume that the mapping is onto, we note that (1) The map must split. (2) If the row is (a1, ..., an), the elements ai must generate the unit ideal in R. This is because the image of the epimorphism is the ideal generated by the ai’s and it must contain 1. Let us call such rows “unimodular rows”. Now consider the kernel of the epimorphism obtained from a unimodular row. Clearly the kernel is a projective module P such that P ⊕ R = R. We would like to obtain conditions under which P is a free module. Since R is assumed to be commutative (and hence R has invariant basis number), this is equivalent to asking whether P ≈ Rn−1. It is easy to see that the above condition is equivalent to the unimodular row being completable to an invertible matrix. For a proof see [7]. Defintion 1.1 :Stable Range: Let {a1, ..., an} be a set of elements generating the unit ideal. The set is said to be reducible if there exist r1, ..., rn−1 such that a1 + r1an, ..., an−1 + rn−1an still generate the unit ideal. If m is such that every such sequence of length exceeding m is reducible, the ring R is said to have stable range m. Let us denote “stable range” by sr. Definition 1.2 :Unimodular Elements: Let M be an R module. Given any x ∈ M , we define the ideal oM (x) = {f(x)∀f ∈ Hom(M,R)}. x is said to be a unimodular element if oM (x) = R. Therefore, by a unimodular row, we shall mean a row a1, ..., an, the ideal generated by which is R. We shall first show that “long” unimodular rows over a ring are always completable.