symmetric monoidal (∞,1)-category of spectra
A (left/right/2-sided) principal ideal in a ring is a left/right/2-sided ideal generated by an element , or equivalently a left sub--module/right sub--module/sub---bimodule generated by .
This means there exists an element such that is a multiple of whenever ; we say that is generated by . Thus every element generates a unique principal ideal, the set of all left/right/two-sided multiples of : , , or if we are talking about left/right/two-sided ideals in a ring. Clearly, every ideal is a join over all the principal ideals generated by the elements of .
In commutative rings, since the set of all principal ideals is isomorphic to the quotient of (the multiplicative monoid structure on) by the group of units, a principal ideal is equivalently an element of the quotient monoid .
Last revised on January 23, 2023 at 18:40:53. See the history of this page for a list of all contributions to it.