“commutative algebra”的概念、定义、翻译、参考文献-科学参考

commutative algebra

Mathematics

The branch of algebra concerned with commutative rings, which provides the rigour and techniques for algebraic geometry. Varieties are defined as the zero sets of polynomials to which commutative rings can be associated. The geometric properties of the varieties (such as singular points) can be determined from the study of the associated rings. For example, the ring associated with the curve y² = x² in ℂ² is

$\frac{ℂ [x, y]}{〈 x^{2} - y^{2} 〉} ≅ \frac{ℂ [x, y]}{〈 x + y 〉} \oplus \frac{ℂ [x, y]}{〈 x - y 〉} .$

Note that 〈x + y〉 and 〈x−y〉 are prime ideals of ℂ[x,y] and correspond to the irreducible components x + y = 0 and x−y = 0 of the curve. See also noetherian ring.

单词	commutative algebra
释义	commutative algebra Mathematics The branch of algebra concerned with commutative rings, which provides the rigour and techniques for algebraic geometry. Varieties are defined as the zero sets of polynomials to which commutative rings can be associated. The geometric properties of the varieties (such as singular points) can be determined from the study of the associated rings. For example, the ring associated with the curve y² = x² in ℂ² is $\frac{ℂ [x, y]}{〈 x^{2} - y^{2} 〉} ≅ \frac{ℂ [x, y]}{〈 x + y 〉} \oplus \frac{ℂ [x, y]}{〈 x - y 〉} .$ Note that 〈x + y〉 and 〈x−y〉 are prime ideals of ℂ[x,y] and correspond to the irreducible components x + y = 0 and x−y = 0 of the curve. See also noetherian ring.
随便看	feral animal Ferdinand Ferdinand II (1578–1637) Ferdinand V (1452–1516) Ferdinand VII (1784–1833) Ferguson, Adam (1723–1816) Fermat, Pierre de Fermat, Pierre de (1601–65) Fermat, Pierre de (1607–65) Fermat point Fermat prime Fermat’s Last Theorem Fermat’s Little Theorem Fermat’s principle Fermat’s Theorem Fermat’s Two Squares Theorem fermentation fermenter fermi Fermi acceleration Fermi constant Fermi energy Fermi, Enrico Fermi, Enrico (1901–54) Fermi Gamma-Ray Space Telescope