See integral domain on Wiktionary
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"text": "A ring R is an integral domain if and only if the polynomial ring R#x5B;x#x5D; is an integral domain.",
"type": "example"
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"text": "For any integral domain there can be derived an associated field of fractions.",
"type": "example"
},
{
"ref": "1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266:",
"text": "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).",
"type": "quote"
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"ref": "2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95:",
"text": "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.",
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"ref": "2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171:",
"text": "#x5B;#x5C;mathbb#x7B;Z#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, #x5B;#x5C;mathbb#x7B;Z#x7D;#x5F;p,#x2B;#x5F;p,#x5C;times#x5F;p#x5D; with p a prime, #x5B;#x5C;mathbb#x7B;Q#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, #x5B;#x5C;mathbb#x7B;R#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, and #x5B;#x5C;mathbb#x7B;C#x7D;#x3B;#x2B;,#x5C;cdot#x5D; are all integral domains. The key example of an infinite integral domain is #x5B;#x5C;mathbb#x7B;Z#x7D;#x3B;#x2B;,#x5C;cdot#x5D;. In fact, it is from #x5C;mathbb#x7B;Z#x7D; that the term integral domain is derived. Our main example of a finite integral domain is #x5B;#x5C;mathbb#x7B;Z#x7D;#x5F;p,#x2B;#x5F;p,#x5C;times#x5F;p#x5D;, when p is prime.",
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"Any nonzero commutative ring in which the product of nonzero elements is nonzero."
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"word": "field of fractions"
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"word": "commutative ring"
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"id": "en-integral_domain-en-noun-OL98CWCr",
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"(algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero."
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"word": "entire ring"
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"translations": [
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"code": "fi",
"lang": "Finnish",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"word": "kokonaisalue"
},
{
"code": "fr",
"lang": "French",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
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"masculine"
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"word": "anneau d’intégrité"
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{
"code": "it",
"lang": "Italian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
"tags": [
"masculine"
],
"word": "dominio d'integrità"
},
{
"code": "sh",
"lang": "Serbo-Croatian",
"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
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"Cyrillic"
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"word": "област цијелих"
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"word": "domen"
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"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
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"word": "dominio de integridad"
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"word": "integral domain"
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"text": "A ring R is an integral domain if and only if the polynomial ring R#x5B;x#x5D; is an integral domain.",
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"text": "For any integral domain there can be derived an associated field of fractions.",
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"ref": "1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266:",
"text": "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).",
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"ref": "2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95:",
"text": "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.",
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"ref": "2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171:",
"text": "#x5B;#x5C;mathbb#x7B;Z#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, #x5B;#x5C;mathbb#x7B;Z#x7D;#x5F;p,#x2B;#x5F;p,#x5C;times#x5F;p#x5D; with p a prime, #x5B;#x5C;mathbb#x7B;Q#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, #x5B;#x5C;mathbb#x7B;R#x7D;#x3B;#x2B;,#x5C;cdot#x5D;, and #x5B;#x5C;mathbb#x7B;C#x7D;#x3B;#x2B;,#x5C;cdot#x5D; are all integral domains. The key example of an infinite integral domain is #x5B;#x5C;mathbb#x7B;Z#x7D;#x3B;#x2B;,#x5C;cdot#x5D;. In fact, it is from #x5C;mathbb#x7B;Z#x7D; that the term integral domain is derived. Our main example of a finite integral domain is #x5B;#x5C;mathbb#x7B;Z#x7D;#x5F;p,#x2B;#x5F;p,#x5C;times#x5F;p#x5D;, when p is prime.",
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"Any nonzero commutative ring in which the product of nonzero elements is nonzero."
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"(algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero."
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"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
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"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
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"word": "anneau d’intégrité"
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"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
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"word": "dominio d'integrità"
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"sense": "nonzero commutative ring in which the product of nonzero elements is nonzero",
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"word": "dominio de integridad"
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"word": "integral domain"
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