_:vb7494463 "For integral domains, we will use a-1 to designate the multiplicative inverse of a (if it has one; since not all elements need have inverses, this notation can be used only where it can be shown that an inverse exists)."@en . _:vb7494464 "An integral domain is said to have strong pseudo-Dedekind factorization if each proper ideal can be factored as the product of an invertible ideal (possibly equal to the ring) and a finite product of pairwise comaximal prime ideals with at least one prime in the product."@en . . . . _:vb7494465 "[\\mathbb{Z};+,\\cdot], [\\mathbb{Z}_p,+_p ,\\times_p] with p a prime, [\\mathbb{Q};+,\\cdot], [\\mathbb{R};+,\\cdot], and [\\mathbb{C};+,\\cdot] are all integral domains. The key example of an infinite integral domain is [\\mathbb{Z};+,\\cdot]. In fact, it is from \\mathbb{Z} that the term integral domain is derived. Our main example of a finite integral domain is [\\mathbb{Z}_p ,+_p ,\\times_p], when p is prime."@en . . . . . _:vb7494463 "1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266:"@en . . _:vb7494460 "(algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero. [from 1911]"@en . _:vb7494464 . . _:vb7494465 . . "1" . _:vb7494461 . _:vb7494460 . _:vb7494464 "2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95:"@en . _:vb7494462 . _:vb7494463 . _:vb7494465 "2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171:"@en . _:vb7494462 "For any integral domain there can be derived an associated field of fractions."@en . _:vb7494461 "A ring R is an integral domain if and only if the polynomial ring R[x] is an integral domain."@en .