Ideal

Principal ideal

A (left/right/two-sided) ideal 𝐼 βŠ΄π‘… is called principal iff it is generated by a single element. ring Specifically, 𝐼 is a

  • left principal ideal iff 𝐼 =π‘…π‘Ž for some π‘Ž βˆˆπ‘…;
  • right principal ideal iff 𝐼 =π‘Žπ‘… for some π‘Ž βˆˆπ‘…;
  • two-sided principal ideal iff 𝐼 =π‘…π‘Žπ‘… for some π‘Ž βˆˆπ‘….

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

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