Editing Principal ideal (section)

==Definitions==
* A ''left principal ideal'' of <math>R</math> is a [[subset]] of <math>R</math> given by <math>Ra = \{ra : r \in R\}</math> for some element <math>a.</math>
* A ''right principal ideal'' of <math>R</math> is a subset of <math>R</math> given by <math>aR = \{ar : r \in R\}</math> for some element <math>a.</math>
* A ''two-sided principal ideal'' of <math>R</math> is a subset of <math>R</math> given by <math>RaR = \{r_1 a s_1 + \ldots + r_n a s_n: r_1,s_1, \ldots, r_n, s_n \in R\}</math> for some element <math>a,</math> namely, the set of all finite sums of elements of the form <math>ras.</math>

While the definition for two-sided principal ideal may seem more complicated than for the one-sided principal ideals, it is necessary to ensure that the ideal remains closed under addition.{{r|n=df3ed|pp=251-252|r={{cite book|last=Dummit|first=David S.|last2=Foote|first2=Richard M.|title=Abstract Algebra|edition=3rd|publisher=John Wiley & Sons|publication-place=New York|date=2003-07-14|isbn=0-471-43334-9}}}}

If <math>R</math> is a [[commutative ring]], then the above three notions are all the same. In that case, it is common to write the ideal generated by <math>a</math> as <math>\langle a \rangle</math> or <math>(a).</math>