See commutative ring on Wiktionary
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"ref": "1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129:",
"text": "Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications.",
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"ref": "2002, Joseph J. Rotman, Advanced Modern Algebra, 2nd edition, American Mathematical Society, page 295:",
"text": "As usual, it is simpler to begin by looking at a more general setting—in this case, commutative rings—before getting involved with polynomial rings. It turns out that the nature of the ideals in a commutative ring is important: for example, we have already seen that gcd's exist in PIDs, while this may not be true in other commutative rings.",
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"ref": "2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, page 47:",
"text": "In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#x5C;inP, it follows that a#x5C;inP or b#x5C;inP; equivalently, P is a prime ideal if and only if the factor ring R#x2F;P is a domain.",
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"A ring whose multiplicative operation is commutative."
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"sense": "a ring whose multiplicative operation is commutative",
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"word": "anello commutativo"
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"alt": "かかんかん",
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"word": "commutative ring"
}
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This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-11-06 from the enwiktionary dump dated 2024-10-02 using wiktextract (fbeafe8 and 7f03c9b). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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