Lecture Notes for Math 627A Modern Algebra Notes on the Isomorphism Theorems for Groups

Lecture Notes for Math 627B Modern Algebra Notes on the Isomorphism Theorems for Groups

Lecture Notes for Math 627A Modern Algebra Modules and Groebner Bases

Throughout this document any ring is a commutative ring with identity. Any time we use k it is a field. The theory of modules over a ring is a generalization of the linear algebra, the theory of vector spaces over a field. Many of the terms and results for vector spaces extend quite easily to modules, but there are some issues that require careful formulation. The theory of modules is also a ge...

Some notes on the existence of an inequality in Banach algebra

We shall prove an existence inequality for two maps on Banach algebra, withan example and in sequel we have some results on R and Rn spaces. This waycan be applied for generalization of some subjects of mathematics in teachingwhich how we can extend a math problem to higher level.

Math 627 A : Modern Algebra I Notes on the Isomorphism Theorems for Groups 1 Fundamental Results : Up to the First Isomorphism Theorem

A brief comment about items (3) and (4). If H is a subgroup of G the inclusion map is an injective homomorphism from H to G . On the other hand, the image of an injective homomorphism α : H −→ G is a subgroup that is isomorphic to H . So the study of injective homomorphisms is (up to isomorphism) the study of subgroups. After studying subgroups, we defined normal subgroup and showed several equ...

Math 110: Linear Algebra Practice Midterm #2

∗Note The theorems in sections 5.1 and 5.2 each have two versions, one stated in terms of linear operators, one in terms of matrices. The book states most of them in terms of linear operators, whilst in the lecture notes, they are mostly stated in terms of matrices. For example, compare Theorem 5.5 and its corollary in the book with Theorem 5 and its corollary in the lecture notes; also compare...