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Section Daily Prep 13
You will learn the definition of a ring as well as some basic ring properties and practice instantiating them.
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, commutative ring, unity/identity, unit
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 12, pp. 245-246
Figure 134. Reference Video for Ring Basics
Subsection Important Terms
Definition 135 . Ring.
A ring \(R\) is a set with two binary operations, addition (\(a+b\) ) and multiplication (\(ab\) ), such that for all \(a,b,c\) in \(R\text{:}\)
Addition is commutative:
\(a+b=b+a\)
Addition is associative:
\((a+b)+c=a+(b+c)\)
There is an additive identity
\(0\) such that
\(a+0=a\)
Additive inverses exist: for each
\(a\) there is a
\(-a\) such that
\(a+(-a)=0\)
Multiplication is associative:
\(a(bc)=(ab)c\)
Multiplication distributes over addition:
\(a(b+c)=ab+ac\) and
\((b+c)a=ba+ca\)
A ring is commutative if multiplication is commutative: \(ab=ba\text{.}\) A ring has a unity or identity if there exists an element \(1\in R\) such that \(a1=1a=a\) for all \(a\in R\text{.}\) An element \(a\in R\) is a unit if there exists an element \(b\in R\) such that \(ab=ba=1\text{.}\)
