> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/400-abstract-algebra/algebras/ring.md). # Ring ) A ring consists of a set equipped with 2 binary operations that generalize addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions. A **ring** is an *abelian group* with a second binary operation that is associative, distributive over the abelian group operation, and has the identity element (although the identity is not required by some authors). By extension from the integers, the abelian group operation is called *addition* and the second binary operation is called *multiplication*. Commutativity of a ring's second operation has profound impact on its behaviour, promoting *commutative ring theory* (commutative algebra) as a key topic in ring theory. Examples of *commutative rings*: * the set of integers equipped with the addition and multiplication * the set of polynomials equipped with their addition and multiplication * the coordinate ring of an affine algebraic variety * the ring of integers of a number field Examples of *noncommutative rings*: * the ring of n × n real square matrices with n ≥ 2 * group rings in representation theory * operator algebras in functional analysis * rings of differential operators in the theory of differential operators * the cohomology ring of a topological space in topology --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/400-abstract-algebra/algebras/ring.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.