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Section Daily Prep 16
You will become familiar with the definition and basic examples of a ring homomorphism.
Objectives: Basic Learning Objectives
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
State and instantiate the definition of: ring homomorphism, ring isomorphism, kernel
Given a map between rings, determine whether it is a ring homomorphism and compute its kernel
Determine whether two rings are isomorphic
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 15, pp. 285-287
Figure 161. Reference Video for Ring Homomorphisms
Subsection Important Terms
Definition 162 . Ring Homomorphism.
A ring homomorphism from a ring \(R\) to a ring \(S\) is a map \(\phi: R \to S\) that preserves ring operations: for all \(a, b \in R\text{:}\)
\(\displaystyle \phi(a + b) = \phi(a) + \phi(b)\)
\(\displaystyle \phi(ab) = \phi(a)\phi(b)\)
A ring homomorphism is an isomorphism if it is 1-to-1 and onto.
