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In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers.
In Ring, the category of rings with unity and unity-preserving morphisms, the ring of integers Z is an initial object. The zero ring consisting only of a single element 0 = 1 is a terminal object. In Rig, the category of rigs with unity and unity-preserving morphisms, the rig of natural numbers N is an initial object.
The ring of integers Z is an initial object in Ring. The zero ring is a terminal object in Ring. The product in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise.
• Involutive ring • Category of rings • Initial ring • Terminal ring = / Related structures • ... In algebra, ring theory is the study of rings, ...
For every ring R, there is a unique ring homomorphism Z → R. This says that the ring of integers is an initial object in the category of rings. For every ring R, there is a unique ring homomorphism from R to the zero ring. This says that the zero ring is a terminal object in the category of rings.
• 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