Product of rings

In

componentwise operations. It is a direct product in the category of rings

Since direct products are defined up to an isomorphism, one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings. For example, the Chinese remainder theorem may be stated as: if $m$ and $n$ are coprime integers, the quotient ring $\mathbb {Z} /mn\mathbb {Z}$ is the product of $\mathbb {Z} /m\mathbb {Z}$ and $\mathbb {Z} /n\mathbb {Z} .$

Examples

An important example is Z/nZ, the

ring of integers modulo n. If n is written as a product of prime powers (see Fundamental theorem of arithmetic

),

n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots \ p_{k}^{n_{k}},

where the p_i are distinct

isomorphic

to the product

\mathbf {Z} /p_{1}^{n_{1}}\mathbf {Z} \ \times \ \mathbf {Z} /p_{2}^{n_{2}}\mathbf {Z} \ \times \ \cdots \ \times \ \mathbf {Z} /p_{k}^{n_{k}}\mathbf {Z} .

This follows from the Chinese remainder theorem.

Properties

If R = Π_i∈I R_i is a product of rings, then for every i in I we have a

surjective ring homomorphism p_i : R → R_i which projects the product on the i th coordinate. The product R together with the projections p_i has the following universal property

:

if S is any ring and f_i : S → R_i is a ring homomorphism for every i in I, then there exists precisely one ring homomorphism f : S → R such that p_i ∘ f = f_i for every i in I.

This shows that the product of rings is an instance of products in the sense of category theory.

When I is finite, the underlying additive group of Π_i∈I R_i coincides with the

trivial

, the inclusion map R_i → R fails to map 1 to 1 and hence is not a ring homomorphism.

(A finite coproduct in the

free product of algebras

.)

Direct products are commutative and associative up to natural isomorphism, meaning that it doesn't matter in which order one forms the direct product.

If A_i is an

a fortiori

prime.

An element x in R is a

group of units of R is the product

of the groups of units of the R_i.

A product of two or more non-trivial rings always has nonzero

zero divisors

: if x is an element of the product whose coordinates are all zero except p_i (x) and y is an element of the product with all coordinates zero except p_j (y) where i ≠ j, then xy = 0 in the product ring.

References

ISBN 978-0-88385-039-8

Zbl 0984.00001

Algebraic structure → Ring theory Ring theory
Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring $\mathbb {Z}$ • Terminal ring $0=\mathbb {Z} /1\mathbb {Z}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield
Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring $\mathbb {Z} /1\mathbb {Z}$ • Integers modulo pⁿ $\mathbb {Z} /p^{n}\mathbb {Z}$ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$ • Base-p circle ring $\mathbb {T}$ • integers $\mathbb {Z}$ • p-adic rationals $\mathbb {Z} [1/p]$ • Base-p real numbers $\mathbb {R}$ • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • p-adic solenoid $\mathbb {T} _{p}$ Algebraic geometry • Affine variety
Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra
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