Definition
A noncommutative ring is an algebraic structure $(R, +, \cdot)$ consisting of a set $R$ equipped with two binary operations: addition (+) and multiplication (·). The set $R$ forms an abelian group under addition, multiplication is associative, and multiplication distributes over addition from both sides. Unlike a commutative ring, a noncommutative ring does not, in general, satisfy the commutative law $a \cdot b = b \cdot a$ for all elements $a, b \in R$.
Overview
Noncommutative rings arise naturally in many areas of mathematics, including abstract algebra, functional analysis, and representation theory. Classic examples include matrix rings $M_n(F)$ of $n \times n$ matrices over a field $F$ (for $n \ge 2$), the quaternions $\mathbb{H}$, and group rings of non‑abelian groups. The study of these rings extends many concepts from commutative algebra—such as ideals, modules, and homological dimensions—while also introducing phenomena absent in the commutative setting, such as one‑sided ideals and the failure of the cancellation property for multiplication.
Etymology / Origin
The term combines “non‑commutative,” meaning “not commutative,” with “ring,” a term introduced in the early 20th century by mathematicians such as Emmy Noether and David Hilbert to denote a set with two compatible operations. The adjective “non‑commutative” was adopted to distinguish rings that do not satisfy the commutative law for multiplication, a distinction that became essential as algebraic structures beyond the integers and polynomial rings were systematically investigated.
Characteristics
| Property | Description |
|---|---|
| Additive structure | $(R, +)$ is an abelian (commutative) group; there exists an additive identity $0$ and additive inverses. |
| Multiplicative associativity | For all $a, b, c \in R$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. |
| Distributivity | Multiplication distributes over addition: $a\cdot(b + c) = a\cdot b + a\cdot c$ and $(a + b)\cdot c = a\cdot c + b\cdot c$. |
| Lack of commutativity | There exist elements $a, b \in R$ such that $a \cdot b |
| eq b \cdot a$. | |
| Identity element (optional) | Many authors require a multiplicative identity $1$ with $1\cdot a = a\cdot 1 = a$; rings without such an element are called rngs. |
| Ideals | Both left ideals ${x \in R \mid rx \in I\ \forall r \in R}$ and right ideals ${x \in R \mid xr \in I\ \forall r \in R}$ are distinguished; two‑sided ideals satisfy both conditions. |
| Modules | Generalizations of vector spaces; a (left) $R$-module $M$ allows scalar multiplication $R \times M \to M$ respecting the ring operations. |
| Center | The subset $Z(R) = {z \in R \mid zr = rz\ \forall r \in R}$ forms a commutative subring; it measures the degree of non‑commutativity. |
| Division rings | A noncommutative ring in which every non‑zero element has a multiplicative inverse; also called skew fields. |
Related Topics
- Commutative ring – a ring where multiplication is commutative.
- Division ring (skew field) – a noncommutative ring with multiplicative inverses for all non‑zero elements.
- Matrix ring – the ring of $n \times n$ matrices over a field or another ring; a primary example of a noncommutative ring for $n \ge 2$.
- Quaternion algebra – a four‑dimensional noncommutative division algebra over $\mathbb{R}$.
- Group ring – constructed from a group $G$ and a ring $R$; noncommutative when $G$ is non‑abelian.
- Ring theory – the broader study of rings, encompassing both commutative and noncommutative cases.
- Module theory – the study of modules over rings, central to understanding representations of noncommutative rings.
- Noncommutative geometry – a field extending geometric concepts to settings where coordinate algebras are noncommutative rings.