“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	Aitken, Robert Grant (1864–1951) Aitken’s method(in numerical analysis) Aitken’s Δ2 process AIUI AIV AI Velorum star AIW AIX Aix‐la‐Chapelle, Treaty of (1748) AJ Ajax Ajdukiewicz, Kazimierz (1890–1963) Ajne test AKA Akaike, Hirotugu (1927–2009) Akaike's information criterion Akari Akatsuki Akbar, Jalaludin Muhammad (1542–1605) Akebono Akers, Thomas Dale (1951– ) Akhenaten (14th century bc) akinete Akiyama, Toyohiro (1942– ) Akkad