By Randall R. Holmes

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**Additional resources for Abstract Algebra II**

**Sample text**

So here is a second characterization: • a prime integer is an integer p, neither zero nor a unit, having the property that if p|mn (m, n ∈ Z), then either p|m or p|n. These two characterizations of prime integer are equivalent and they describe the usual prime numbers together with their negatives: ±2, ±3, ±5, ±7, ±11, and so on. We alert the reader to a point of terminology. The natural generalizations of these two characterizations to other integral domains are not always equivalent, so in general one cannot use the same term for both properties as we have done here with the word “prime”.

Let r = st (s, t ∈ R) be a factorization of r and assume that s is a nonunit. Then (s) is proper and r ∈ (s), which implies that (r) ⊆ (s). By maximality of (r), we have (r) = (s), so that r ∼ s. This implies that r = su for some unit u ∈ R. Therefore, st = su and by cancellation, t = u, so that t is a unit. (ii) Assume that r is prime. Arguing as above, (r) is nonzero and proper. Let s, t ∈ R. Assume that st ∈ (r) and s ∈ / (r). Then r | st and r s, implying that r | t so that t ∈ (r). Now assume that (r) is prime.

In more detail, V /W = {v + W | v ∈ V } is a vector space with addition and scalar multiplication given by (i) (v + W ) + (v + W ) = (v + v ) + W , (ii) a(v + W ) = (av) + W . These operations are well-defined, meaning that they do not depend on the choices of coset representatives. V /W is the quotient of V by W . 6 Span Let F be a field, let V be a vector space over F , and let S be a subset of V . The span of S, written S , is the intersection of all subspaces of V that contain S: W. S = W ≤V W ⊇S Since an intersection of subspaces is again a subspace, the span of S is a subspace of V .

### Abstract Algebra II by Randall R. Holmes

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