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 ...

... 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 ...

Commutative Algebra

... 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 ...

... 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. ...

... 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. ...

Divided powers

... (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 ...

... (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 ...

... 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 ...