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.
Now you Không máy tính nào có thể tính toán được nó, nhưng các mảnh của nó có thể tính toán được. Thử một trong những cách dưới đây, hoặc gõ theo ý bạn.
Ký hiệu được dùng ở đây
Nhấn vào bất kỳ ký hiệu nào để xem định nghĩa đầy đủ, một bức ảnh, và ý nghĩa của từng chữ cái trong nó.
Câu hỏi người ta hỏi
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.
Một số phần của trang này được chuyển từ Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Được nén lại và giải thích lại ở đây; lỗi là của chúng tôi.
More in 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