# coprime vs relatively prime

Since I am not a native english speaker and really couldn't find it: Which definition in a ring (of course not integers) belongs to which word:

Let $R$ be a ring (commutative with 1) and $a,b \in R$, then $a,b$ are called relatively prime resp. coprime if

1. For any $c \in R: c$ divides $a$ and $b \implies c \in R^\times$

or

1. The ideal generated for $a$ and $b$: $\,(a,b)$ is $R$.

Of course these two definitions coincide in PIDs but in other rings they obviously differ. The question is now, which is which?

An easy example where it differs would be the polynomial ring over 2 variables $R=K[X,Y]$ for some field $K$ and $a=X, b=Y$. They fullfill 1. but not 2.

• May I ask you where did you find those definitions? Usually the theory of divisibility is given in the context of integral domains.
– Xam
Commented Nov 9, 2017 at 1:52
• I've usually heard the term comaximal for your second definition. Ideals $I$ and $J$ are comaximal if $I + J = R$. So in your example of $R = K[x,y]$, the elements $x$ and $y$ are coprime, but not comaximal. Commented Nov 9, 2017 at 3:49
• Btw, your second definition implies the first one. Indeed, if $c\mid a.b$ then $(a,b)\subseteq (c)$, so $R=(c)$, which implies that $c\in R^\times$.
– Xam
Commented Nov 9, 2017 at 3:51
• @Xam I currently assist a course which is given in english, but our main source is a german book, where there is no verbal distinction between coprime and relatively prime, i.e. the author states directly which definition he means. But since we give the lecture in english, I would like to know, which term would be correct (or if there is a distinction at all). I have seen and used both definitions on several occasions (mostly the first one), but can't recall properly which is which, since mostly its clear from the context or they are equivalent anyway. In talking I don't mind using the wrong
– ctst
Commented Nov 9, 2017 at 14:40
• word, since I assume my speaking partner knows, what I mean, but in the course I think it would be nice to use the correct english term.
– ctst
Commented Nov 9, 2017 at 14:41

### Elements satisfying $(a,b)=R$.

The following authors refer to elements satisfying this condition as relatively prime:

• N. Bourbaki, see Commutative Algebra Chapter II, § 1.2
• S. Lang, see Algebra, see Chapter II, § 5

But the following authors refer to elements satisfying this same condition as coprime or comaximal:

• M.F. Atiyah & I.G. MacDonald, see Introduction to Commutative Algebra, top of page 7

### Elements with no common factor

I have trouble finding any terminology for this in the literature, but I am fairly certain that I have seen the terminology coprime (also) used to indicate this condition.

I invite anybody with further references to add them to the above list.