In general, a ring can be thought of as a set[1] that has been combined with two binary operations[2] and follows certain rules regarding what happens when various elements of the set are combined with each other using the two binary operations.
More specifically, to get a ring, start with an Abelian group[3] whose operation is called addition. Now, pair it with a second binary operation called multiplication. This second binary operation, multiplication, must be associative[4] and must be distributive[5] over the addition operation. The Wikipedia article “Ring (mathematics)” says that multiplication must also have an identity[6], but acknowledges that some authors do not require rings to have a multiplicative identity. For example, Abstract Algebra: Theory and Applications, by Thomas W. Judson, specifically calls a ring with a multiplicative identity a “ring with unity” or a “ring… with identity.”
The addition operation is often denoted with a plus sign (“+”) while the multiplication operation is often denoted with a multiplication sign (“*”) or no sign.
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Sources: “Ring (mathematics),” Wikipedia, retrieved 12/15/2020, and Abstract Algebra: Theory and Applications, by Thomas W. Judson, August 9, 2016 edition.
- [1] See “What is the Gist of a “Set” (Mathematics)?”
- [2] See “What is the Gist of a “Binary Operation”?”
- [3] See “What is the Gist of an “Abelian Group”?”
- [4] See “What is the Gist of “Associativity” (Mathematics)?”
- [5] See “What is the Gist of the “Distributive Property” (Mathematics)?”
- [6] See “What is the Gist of “Identity” (Mathematics)?”
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Disclaimer:
I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).
