The number theoretic functions equipped with Dirichlet convolution and prefix sum, form one of the most important topics in elementary number theory.In this text, we are going to study the number theoretic functions, however this is done from algebraic perspective by studying the Dirichlet ring, which they form under addition and Dirichlet convolution.Actually, one who gets the chance to study both elementary number theory and algebra in parallel will find how convenience and simple it is by...(read more)

We have already been familiar with the fundamental theorem of arithmetic every since our elementary school, that for every positive integer can be decomposed into product of postitive prime power form , and the decomposition is unique.When it comes to the case of decomposing a polynomial in over , for a reduction of polynomial , it...(read more)

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)