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 general rings." (my emphasis)
Then, the first sentence on page 274 reads as follows: (see attachment - page 274)
"Definition. Let R be a commutative ring and let with
... ... "
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 if and only if ."
My problem is this - I think D&F should have defined R as a commutative ring with identity since proving that requires the ring to have an (multiplicative) identity or unity.
Can someone please confirm or clarify this for me?