We have already got to know the fractions in our elementary school: first we get used to the operations of integers , and found none of them except for has its multiplicative inverse, such that multiplying the inverse is equivalent to dividing them; to mitigate, we introduce the concept of fractions and rational numbers , so that division is free and the operation in the original ring is...(read more)

We've actually informally introduced and discussed about vector spaces when we were proving the Wedderburn's little theorem, stating that finite division rings are vector spaces over their centers. We are using vector space as a tool for deriving numberical relationship between the elements in the division ring and and the elements in the center.Vector spaces and linear transformations are concepts from undergraduate linear algebra, but in this text, we will extend them to ring theory discuss them in...(read more)

We have already been familiar with polynomials like or defined on numbers. Actually the concept of polynomials can be extended to rings, where the polynomials also forms a ring and can be studied as a topic of ring theory.In this text we will first introduce and study the ring properties of polynomials; then we will study the factorization of irreducibility of polynomials over fields; we prove the...(read more)

Centered at the study of algebra is the study of homomorphisms between algebraic structures. For the case of ring, a ring homomorphisms is connected by a special kind of subring called the ideal, which is the kernel of the homomorphism and has absorption property.In this text, we will first compare and migrate the isomorphism theorems into ring theory, and then categorize the ideals into principal, maximal and prime ideals. This should establish...(read more)

Ring theory is namely a branch of algebra that studies a type of algebraic structure called ring. A ring is built upon an abelian group defining the addition, and adds multiplication which is another binary operation that is closed, commutative and distributive over the addition. The multiplication is not limited to something originated from repetive sum of the addition, things like convolution that has distributive properties can be used in the place...(read more)