integral domain

integral domain (plural integral domains)

1. (algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero. [from 1911] [1]

   A ring ${\displaystyle R}$ is an integral domain if and only if the polynomial ring ${\displaystyle R[x]}$ is an integral domain.

   For any integral domain there can be derived an associated field of fractions.

   • 1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266,

     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).

   • 2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95,

     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.

   • 2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171,

     ${\displaystyle [\mathbb {Z} ;+,\cdot ]}$ , ${\displaystyle [\mathbb {Z} _{p},+_{p},\times _{p}]}$ with ${\displaystyle p}$ a prime, ${\displaystyle [\mathbb {Q} ;+,\cdot ]}$ , ${\displaystyle [\mathbb {R} ;+,\cdot ]}$ , and ${\displaystyle [\mathbb {C} ;+,\cdot ]}$ are all integral domains. The key example of an infinite integral domain is ${\displaystyle [\mathbb {Z} ;+,\cdot ]}$ . In fact, it is from ${\displaystyle \mathbb {Z} }$ that the term integral domain is derived. Our main example of a finite integral domain is ${\displaystyle [\mathbb {Z} _{p},+_{p},\times _{p}]}$ , when ${\displaystyle p}$ is prime.

For a list of several equivalent definitions, see Integral domain#Definition on Wikipedia.Wikipedia

• (commutative ring in which the product of nonzero elements is nonzero): entire ring

• commutative ring
• domain

• unique factorization domain, Noetherian domain
  • principal ideal domain
    • Euclidean domain
      • field

• field of fractions

• Zero-product property on Wikipedia.Wikipedia
• Dedekind–Hasse norm on Wikipedia.Wikipedia