# What are S-integral matrices for S a finite set of primes in a number field?

I am reading the paper Geometric construction of cohomology for arithmetic groups I by Millson and Raghunathan, and unfortunately I know almost no number theory, so I am confused by the following proposition:

Specifically, what does it mean when they say "Let $$S=\{p_1, \ldots, p_m\}$$ be a finite set of primes of $$E$$ and let $$G(O_S)$$ be the group of $$S$$ integral matrices over $$E$$ that preserve $$Q$$."

1. What is $$O_S$$ exactly? I found the definition of $$O_R$$ for $$R$$ a ring to be be the ring of roots of monic polynomials with integer coefficients. But $$S$$ is obviously not a ring. Is the intent her to look at the subring generated by $$S$$ and find the roots within this subring?

2. What do they mean be a "finite set of primes of $$E$$"? When I look up the definition of a prime element of a ring it says that an element $$p$$ of a ring $$R$$ is prime "if it is nonzero, has no multiplicative inverse and satisfies the following requirement: whenever $$p$$ divides the product $$xy$$ of two elements of $$R$$, it also divides at least one of $$x$$ or $$y$$." But this doesn't make sense in this context since every non-zero element of a field is invertible. Do you think that the authors mean that these are irreducible elements instead? Or is there some other meaning of prime in this context?

• planetmath.org/RingOfSintegers – Qiaochu Yuan Nov 22 '18 at 3:03
• @QiaochuYuan Thanks! I've added a second question regarding terminology. Could you help me with this as well? – ಠ_ಠ Nov 22 '18 at 7:28
• I think "prime of $E$" means a prime element of the ring of integers of $E$. – Qiaochu Yuan Nov 22 '18 at 20:21

Here $$O_S$$ will be the ring of integers of $$E$$ with the inverses of the $$p_i$$ adjoined. That is, $$O_S=O_E[p_1^{-1},\ldots,p_m^{-1}]$$ where $$O_E$$ is the ring of algebraic integers in $$E$$. (This is standard jargon in algebraic number theory.)