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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.
이제 너 계산기는 이것을 해결하지 않지만, 그 조각은 계산 가능합니다. 아래의 하나를 시도하거나 자신의 것을 입력하십시오.
여기서 사용된 기호
기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.
사람들이 묻는 질문
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.
이 페이지의 일부는 다음에서 변경되었습니다. Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). 여기서 압축하고 다시 설명; 오류는 우리의.
에 더 Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula