Ring Theory Solutions
... 13. Useing the pigeonhole principle, prove that if m and n are relatively
prime integers and a and b are any integers, there exist an integer x such
that x ≡ a mod m and x ≡ b mod n. (Hint: Consider the remainders of
a, a + m, a + 2m, . . . , a + (n − 1)m on division by n.)
Solution: Consider the re ...
... polynomials given initially — simply because they are not unique. For example, the variety (a)
above was given as the zero locus of the polynomial x12 + x22 − 1, but it is equally well the zero locus
of (x12 + x22 − 1)2 , or of the two polynomials (x1 − 1)(x12 + x22 − 1) and x2 (x12 + x22 − 1). In o ...
Algebraic Number Theory Brian Osserman
... using the theory of ideal class groups and the analytic class number formula.
These examples together present a strong case that even if one only wishes to
study problems in elementary number theory, it is often natural and important to
consider more general number systems than the integers or Z/nZ. ...
... (3.1) Introduction. Let M be an A-module. We shall in (3.19) show that
the functor from A-algebras to A-modules which maps an A-algebra B to the Amodule homomorphisms !HomA (M, E(B))! from M to E(B) is representable. That
is, there is a A-algebra !Γ(M )!, and for every A-algebra B a canonical biject ...
NOETHERIAN MODULES 1. Introduction In a finite
... analogue of this for modules and submodules is wrong:
(1) A submodule of a finitely generated module need not be finitely generated.
(2) Even if a submodule of a finitely generated module is finitely generated, the minimal
number of generators of the submodule is not bounded above by the minimal num ...
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field K[x1, ..,xn]. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps.Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, andGaussian elimination for linear systems.Gröbner bases were introduced in 1965, together with an algorithm to compute them (Buchberger's algorithm), by Bruno Buchberger in his Ph.D. thesis. He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work.However, the Russian mathematician N. M. Gjunter had introduced a similar notion in 1913, published in various Russian mathematical journals. These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for local rings was developed independently by Heisuke Hironaka in 1964, who named them standard bases.The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras, like Ore algebras.