Ideal

Principal ideal

A (left/right/two-sided) ideal 𝐼 ⊴𝑅 is called principal iff it is generated by a single element. #m/def/ring Specifically, 𝐼 is a

• left principal ideal iff 𝐼 =𝑅𝑎 for some 𝑎 ∈𝑅;
• right principal ideal iff 𝐼 =𝑎𝑅 for some 𝑎 ∈𝑅;
• two-sided principal ideal iff 𝐼 =𝑅𝑎𝑅 for some 𝑎 ∈𝑅.

A ring in which every ideal is (left/right/two-sided) principal is called a (left/right/two-sided) principal ideal ring.

#state/tidy | #lang/en | #SemBr