Here I put some familiar concepts in abstract algebra to help remembering them. Most of the notions and explanations are obtained from Wikipedia. I simplify (may over simplify) the concepts scattered in different places and put them together in a more logic-clear order (as least for myself).

- A
**group**is a set of elements together with an operation that combines two of its elements to form a third element satisfying the**closure**,**associativity**,**identity**and**invertibility**axioms. A group has the most widely applications in Physics, Chemistry, and other fields. The notion of groups arose in the 1830s, and was firmly established around 1870. Modern group theory studies groups in their own right. Various notions are devised to break groups into smaller, better understandable pieces, such as subgroups, quotient groups and other groups. - An
**abelian group**is a group in which the axiom of**commutativity**is added, i.e., the group operation to two group elements does not depend on their order. The abelian group is generally simpler than that of their non-abelian counterparts. Finite abelian groups are very well understood. Infinite abelian groups are an area of current research. - A
**ring**is an abelian group with another binary operation. The abelian group operation is called “addition”, and the newly added operation is called “multiplication” in analogy with the integers. The multiplication is**associative**and**distributive**over the addition (both left and right distributivity). We also agree (or assume) that there is a multiplicative identity `1′ in a ring. If the multiplicative identity does not exist, we call it a**rng**(or a**pseudo-ring**). A ring can be considered as a generalization of the integers to other mathematical objects, such as polynomials, series, matrices and functions. The formal definition of rings is relatively recent, dating from the 1920s. - A
**commutative ring**is a ring in which the**multiplication operation is commutative**. The study of commutative rings is called commutative algebra. - A
**field**is a commutative ring that contains a**multiplicative inverse**for every nonzero element, or equivalently whose nonzero elements form an abelian group under multiplication. Fields are the most familiar concepts in the sense that we learn the knowledge about fields in our primary and middle schools. In a field, we can have the notions of addition, subtraction, multiplication and division. Common fields are the real numbers, the complex numbers, the rational numbers, the functions and so forth.

To summarize:

- groups = the most basic algebraic structures (closure, associativity, identity and invertibility);
- abelian groups = groups + commutativity;
- rings = abelian groups + associative and distributive multiplication;
- commutative rings = rings + commutative multiplication;
- fields = commutative rings + inverse multiplication.

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