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Suppose R and S are rings, and ρ : R → S is a ring homomorphism.
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Morphisms in Rng (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms).
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Show that ψa is a ring homomorphism.
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In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity).
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In addition, ring homomorphisms are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence (Commutative rings)op ≅ (affine schemes) Affine schemes are the local building blocks of schemes.
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The set of all group endomorphisms of M is denoted EndZ(M) and forms a ring under addition and composition, and sending a ring element r of R to its action actually defines a ring homomorphism from R to EndZ(M).
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Such a ring homomorphism R → EndZ(M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it.
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One can show that Cτ(X) = Cρ(X) in the above construction so that the rings C(X) and Cb(X) are typically only studied for completely regular spaces X. The category of real compact Tychonoff spaces is anti-equivalent to the category of the rings C(X) (where X is real compact) together with ring homomorphisms as maps.
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If M and N are two simple modules over a ring R, then any homomorphism f: M → N of R-modules is either invertible or zero.
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This same construction may be carried out if the Ai's are sets, semigroups, topological spaces, rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms are morphisms in the corresponding category.
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These are precisely the ring homomorphisms which are continuous with respect to the given topologies on R and S.
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Back to our example: take the given rng R, and make a category E whose objects are rng homomorphisms R → S, with S a ring having a multiplicative identity.
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Given a ring homomorphism f: R → S {\displaystyle f:R\to S}, the set of all elements mapped to 0 by f is called the kernel of f.
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If (R, m) and (S, n) are local rings, then a local ring homomorphism from R to S is a ring homomorphism f: R → S with the property f(m) ⊆ n.
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The latter emphasized groups, rings, and homomorphisms as the basic concepts.
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In this section objects are understood to consist of underlying sets with a given algebraic structure, such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. With this in mind, homomorphisms are understood in the corresponding setting (group homomorphisms, etc.).
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A fully homomorphic scheme does not require data decryption to operate, but rather operates over the cyphertext ring, keeping an homomorphism between the cyphertext ring and the plaintext ring.
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To perform diagram chasing, we assume that we are in a category of modules over some ring, so that we may speak of elements of the objects in the diagram and think of the morphisms of the diagram as functions (in fact, homomorphisms) acting on those elements.
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In particular, if R is commutative then the collection of R-module homomorphisms h: M → N (see below) is an R-module (and in fact a submodule of NM). If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C∞(X).
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