maths.free › Abstract Algebra › Rings › Ring Homomorphisms and Ideals

Ring Homomorphisms and Ideals

In the study of groups, a homomorphism is a map that preserves the operation of the group. Similarly, a homomorphism between rings preserves the operations of addition and multiplication in the ring.

Ring Homomorphisms and Ideals

In the study of groups, a homomorphism is a map that preserves the operation of the group. Similarly, a homomorphism between rings preserves the operations of addition and multiplication in the ring. More specifically, if \(R\) and \(S\) are rings, then a ring homomorphism is a map \(\phi : R \rightarrow S\) satisfying \[\begin{aligned}\phi( a + b ) & = \phi( a ) + \phi(b) \\ \phi( a b ) & = \phi( a ) \phi(b)\end{aligned}\] for all \(a, b \in R\). If \(\phi : R \rightarrow S\) is a one-to-one and onto homomorphism, then \(\phi\) is called an isomorphism of rings.

The set of elements that a ring homomorphism maps to \(0\) plays a fundamental role in the theory of rings. For any ring homomorphism \(\phi : R \rightarrow S\), we define the kernel of a ring homomorphism to be the set \[\begin{aligned}\end{aligned}\].

Example

For any integer \(n\) we can define a ring homomorphism \(\phi : {\mathbb Z} \rightarrow {\mathbb Z}_n\) by \(a \mapsto a \pmod{n}\). This is indeed a ring homomorphism, since \[\begin{aligned}\phi( a + b ) & = (a + b) \pmod{n} \\ & = a \pmod{n} + b \pmod{n} \\ & = \phi( a ) + \phi(b)\end{aligned}\] and \[\begin{aligned}\phi( a b ) & = ab \pmod{n} \\ & = a \pmod{n}\cdot b \pmod{n} \\ & = \phi( a ) \phi(b)\end{aligned}\]. The kernel of the homomorphism \(\phi\) is \(n {\mathbb Z}\).

Example

Let \(C[a, b]\) be the ring of continuous real-valued functions on an interval \([a,b]\) as in . For a fixed \(\alpha \in [a, b]\), we can define a ring homomorphism \(\phi_{\alpha} : C[a, b] \rightarrow {\mathbb R}\) by \(\phi_{\alpha} (f ) = f( \alpha)\). This is a ring homomorphism since \[\begin{aligned}\phi_{\alpha}( f + g ) = (f + g)( \alpha) = f(\alpha) + g(\alpha) = \phi_{\alpha}( f ) + \phi_{\alpha}(g ) \\ \phi_{\alpha}( f g ) = (f g)( \alpha) = f(\alpha) g(\alpha) = \phi_{\alpha}( f ) \phi_{\alpha}(g )\end{aligned}\]. Ring homomorphisms of the type \(\phi_{\alpha}\) are called evaluation homomorphisms.

In the next proposition we will examine some fundamental properties of ring homomorphisms. The proof of the proposition is left as an exercise.

Example

Every ring \(R\) has at least two ideals, \(\{ 0 \}\) and \(R\). These ideals are called the trivial ideals.

Let \(R\) be a ring with identity and suppose that \(I\) is an ideal in \(R\) such that \(1\) is in \(I\). Since for any \(r \in R\), \(r1 = r \in I\) by the definition of an ideal, \(I = R\).

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.

Tsopano inu Sikuti pali calculator yomwe imatha kufotokoza izi, koma zigawo zake zimatha kuwerengedwa. Timafuna kuyesera imodzi pansipa, kapena tidzalemba ya ife.

Kusunga ntchito yanu yokha

M’malo mwake, maakaunti aulere amawonjezera malemba pazophunzira zonse, mnda wa zomwe mwamaliza, mavuto anu omaliza m’malo limodzi, ndi m’bale amene mungafunse za nkhaniyo. Maphunziro a matekinoloje ndi otsegulira kwa aliyense, olembetsa kapena osalembetsa.

Kulembetsa Kulowa

Zithunzi

Pitani pa dzina lililonse lachifaniziro kuti mudziwe tanthauzo lake, chithunzi chake, ndi zimene limatanthauza.

Mafunso omwe anthu amafunsa

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Zigawo za m'nkhaniyi ndi zochokera Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Kuphatikizapo ndi kufotokozanso pano; zolakwika ndi zathu.

Zambiri pa Abstract Algebra

Groups Subgroups, cosets and Lagrange's theorem Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula