Ring theory studies rings, their ideals, modules, and structure-preserving ring homomorphisms. It includes both commutative algebra and noncommutative algebra. Here rings are associative and have an identity unless another convention is specified.
The integers are a commutative example. Matrix rings are generally noncommutative: in , while . Thus techniques that reorder scalar factors do not automatically work for matrices.
Quotient rings organize the information lost by a ring homomorphism. If , the first isomorphism theorem for rings identifies with the image of . Studying ideals describes such quotients, while studying modules describes how a ring acts by additive transformations. The two viewpoints connect arithmetic examples with matrix and operator examples.
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Ring theory is a branch of abstract algebra that studies algebraic structures known as rings. A ring is a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. Specifically, a ring \( R \) is defined by the following properties: 1. **Addition**: - The set \( R \) is closed under addition.