First order algebraic ODE, differential algebraic approach by Matsuda M.

By Matsuda M.

Show description

Read Online or Download First order algebraic ODE, differential algebraic approach PDF

Best algebra books

Algebra VII: Combinatorial Group Theory Applications to Geometry

From the reports of the 1st printing of this e-book, released as quantity fifty eight of the Encyclopaedia of Mathematical Sciences:". .. This ebook could be very helpful as a reference and consultant to researchers and graduate scholars in algebra and and topology. " Acta Scientiarum Mathematicarum, Ungarn, 1994 ". .

Additional resources for First order algebraic ODE, differential algebraic approach

Example text

Since Mr C (Rb: Ra)= P, and thus M= P. This shows that every nonzero prime ideal of R is maximal; that is R has Krull dimension 1. 55 M- 1 If M is a maximal ideal of R, then by Theorem 35(3), can be generated by two elements. (3) =:> (4). Let I be a nonzero ideal of R, and choose a non- zero element a e I. sion one, Since R is a Noetherian domain of Krull dimen- I/Ra has finite length as an R-module. • , n-1. is a reflexive ideal of R 1 and ExtR{Ra, R) = 0. hypotheses of Theorem 39 are satisfied.

We have a derived exact sequence obtained by applying the functor HomR(·, R) to exact sequence (a): (b) 0 ~I'~ A'~ Now Ext~(I, R) 2 to ExtR(R/I, R). + rank R' = 2. R' --~ Ext~(I, R) is a torsion R-module, since it is isomorphic Hence it follows from (b) that rank A'= rank 11 But then rank A = rank A', and so A is torsionless by Theorem 27. Let J = Im a' in exact sequence (b). J We have just shown that is not zero, and thus it is isomorphic to a non-zero ideal of R. We now have a commutative diagram with exact rows: 0->R~A~>I~O (c) AAl 0-> AI!

THEOREM 28. If A is a torsion-free R-module of finite rank, then A' is a reflexive R-module. Proof. A 1 :A' ..... A"' bethecanonical R-homomorphisms. A induces a homomorphism µ:A"'_,. A'. A, is the identity on A'. A, is a monomorphism. But then A' is a torsionless R-module, and hence by rank A' = rank A'". A' DEFINITIONS. can not be a is an isomorphism. An R-module will be called a universal injective R-module if it is injective and contains a copy of every simple R-module. J } the injective envelope of J.

Download PDF sample

Rated 4.73 of 5 – based on 5 votes