Ring
Undergraduate (abstract algebra)
Consider a set $R$ with two binary operations, addition $+$ and multiplication $\cdot$. The set $R$ is a ring if
- $R$ with $+$ is a commutative group, whose identity element is written $0$,
- multiplication is associative and has an identity element, written $1$, and
- multiplication distributes over addition: $a (b + c) = a b + a c$ and $(a + b) c = a c + b c$ for all $a, b, c \in R$.
Examples
Examples
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The integers $\mathbb{Z}$, with addition and multiplication
High school
The integers, with addition and multiplication, form a ring, because they form a commutative group under addition, multiplication is associative and has the identity $1$, and multiplication distributes over addition.
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The rational numbers $\mathbb{Q}$, with addition and multiplication
High school
The rational numbers, with addition and multiplication, form a ring, because they satisfy the usual rules of arithmetic, including the distributive law.
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The integers modulo $5$, $\setdef{0, 1, 2, 3, 4}$, with addition and multiplication modulo $5$
Undergraduate
The integers modulo $5$, with addition and multiplication modulo $5$, form a ring, because the rules of integer arithmetic still hold after taking remainders.
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The integers modulo $6$, $\setdef{0, 1, 2, 3, 4, 5}$, with addition and multiplication modulo $6$
Undergraduate
The integers modulo $6$, with addition and multiplication modulo $6$, form a ring, because the rules of integer arithmetic still hold after taking remainders.
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The $2 \times 2$ real matrices, with matrix addition, multiplication by real numbers, and matrix multiplication
Undergraduate
The $2 \times 2$ real matrices form a ring, because matrix multiplication is associative, has the identity matrix $I$ as its identity, and distributes over addition.
Non-examples
Things that may be mistaken for examples.
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The nonzero real numbers, with multiplication
High school
The nonzero real numbers, with multiplication, do not form a ring, because a ring needs an addition too, and the nonzero real numbers are not closed under addition: $1 + (-1) = 0$.
Properties
- Multiplication in a ring need not be commutative. A ring whose multiplication is commutative is called a commutative ring.
- Elements of a ring need not have multiplicative inverses, so division need not be possible.
- In every ring, $0 \cdot a = a \cdot 0 = 0$ and $(-1) \cdot a = -a$ for all $a$.
Notes
Some authors do not require a multiplicative identity $1$. Those authors call a ring with $1$ a unital ring, and some call a ring without $1$ a rng. Under that convention, the even integers form a ring; under the convention used here, they do not, because $1$ is not even.
Distinctions involving this term
- Group vs. ring vs. field: Groups, rings, and fields are sets with operations that obey some of the rules of arithmetic. A group has one operation, which is associative, has an identity element, and can be undone. A ring has an addition and a multiplication; its addition forms a commutative group, but its elements need not have multiplicative inverses. A field is a commutative ring in which every nonzero element has a multiplicative inverse, so division by nonzero elements is possible.