Notational Question : (a) = (b)

Question

What does the notation (a) = (b) mean? For context, a and b are elements of a ring. I have tried to find an answer by googling, but it's difficult to do this since I am not sure what to call it.

These two elements of the ring generate the same principle ideal. — vertical.void, Jan 24 '14 at 04:48

score 3 · Answer 1 · answered Jan 24 '14 at 04:31

3

In Algebra,Hungerford which is a standard book in ring theory, (a) is the principal ideal generated by a

answered Jan 24 '14 at 04:31

Lotus

299

1

This notation arises in other texts like Artin and Dummit & Foote as well. – Lost Jan 24 '14 at 04:38

score 2 · Accepted Answer · edited Apr 13 '17 at 12:19

In a ring $\,R\,$ the notation $(a)$ denotes $aR,\,$ the (principal) ideal generated by $\,a.\,$ Thus $(a) = (b)$ means that $aR = bR,\,$ or, equivalently, $\,a\mid b\,$ and $\,b\mid a,\,$ i.e. $\,a\,$ and $\,b\,$ are associates. If, furthermore, $\,a\,$ and $\,b\,$ are cancellable, then this is equivalent to $\,a,b\,$ being strong associates, i.e. $\,a = bu\,$ for some unit $\,u\,$ (but this may fail without cancellability).

Notational Question : (a) = (b)

2 Answers2