Group Theory and Ring Theory

Group theory is an important theory in abstract algebra. A ring is a kind of algebraic system with two operations (addition and multiplication). It has a deep relationship with groups, especially with the Abelian group. In this essay, the ring and the residual class ring will be talked about. Firstly, this passage is aim to talk about some basic knowledge about the ring which will let readers have a basic understanding of a ring. Then this passage will discuss the residual class ring and subring of the residual class ring of modulo. Some concepts about the ring are also mentioned, such as the centre of the ring, the identity of the ring, the classification of a ring, the residual class ring, the field and the zero divisors. The definitions of mathematical terms mentioned before are stated, as well as some examples of the part of those terms are given. In this passage, there are also some lemmas which are the properties of ring and subring. Future studies of rings and subrings can focus on the application of physics.