On page 273 of Dummit and Foote the last sentence reads: (see attachment - page 273)

"The notion of the greatest common divisor of two elements (if it exists)can be made precise in." (my emphasis)general rings

Then, the first sentence on page 274 reads as follows: (see attachment - page 274)

"Definition.Let R be a commutative ring and let $\displaystyle a,b \in R $ with $\displaystyle b \ne 0 $

... ... "

In this definition D&F go on to define multiple, divisor and greatest common divisor in a commutative ring.

D&F then write the following:

"Note that b|a in a ring if and only if $\displaystyle a \in (b) $ if and only if $\displaystyle (a) \subseteq (b) $."

My problem is this - I think D&F should have defined R as a commutative ringsince proving that $\displaystyle (a) \subseteq (b) \longrightarrow a \in (b) $ requires the ring to have an (multiplicative) identity or unity.with identity

Can someone please confirm or clarify this for me?

Peter