Task: Question Answering
Input (English): Context_1 : Karen Ellen Smith (born 1965, Red Bank, New Jersey) is an American mathematician, specializing in commutative algebra and algebraic geometry. She completed her bachelor's degree in mathematics at Princeton University before earning her PhD in mathematics at the University of Michigan in 1993. Currently she is the Keeler Professor of Mathematics at the University of Michigan. In addition to being a researcher in algebraic geometry and commutative algebra, Smith with others wrote the textbook "An Invitation to Algebraic Geometry". Context_2 : In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not or is not required to be commutative. Context_3 : Hossein Zakeri (Persian حسین ذاکری), Prof. Dr. (born 27 December 1942) is an Iranian mathematician. He, along with Prof. R. Y. Sharp, are the founders of generalized fractions, a branch in theory of commutative algebra which expands the concept of fractions in commutative rings by introducing the modules of generalized fractions. This topic later found applications in local cohomology, in the monomial conjecture, and other branches of commutative algebra. Context_4 : Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers formula_1, and "p"-adic integers. Context_5 : Pierre Samuel (12 September 1921 – 23 August 2009) was a French mathematician, known for his work in commutative algebra and its applications to algebraic geometry. The two-volume work "Commutative Algebra" that he wrote with Oscar Zariski is a classic. Other books of his covered projective geometry and algebraic number theory. Context_6 : In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. The name grew out of the central considerations, such as the Lasker–Noether theorem in algebraic geometry, and the ideal class group in algebraic number theory, of the commutative algebra of the first quarter of the twentieth century. It was used in the influential van der Waerden text on abstract algebra from around 1930. Context_7 : Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers formula_1, and p-adic integers. Context_8 : Sir Michael Francis Atiyah ( ; born 22 April 1929) is an English mathematician specialising in geometry. Context_9 : Introduction to Commutative Algebra is a well-known commutative algebra textbook written by Michael Atiyah and Ian G. Macdonald. It deals with elementary concepts of commutative algebra including localization, primary decomposition, integral dependence, Noetherian and Artinian rings and modules, Dedekind rings, completions and a moderate amount of dimension theory. It is notable for being among the shorter English-language introductory textbooks in the subject, relegating a good deal of material to the exercises. Context_10 : In commutative algebra, the Rees algebra of an ideal "I" in a commutative ring "R" is defined to be formula_1The extended Rees algebra of "I" (which some authors refer to as the Rees algebra of "I") is defined asformula_2This construction has special interest in algebraic geometry since the projective scheme defined by the Rees algebra of an ideal in a ring is the blowing-up of the spectrum of the ring along the subscheme defined by the ideal. fact_1 : Introduction to Commutative Algebra is a well-known commutative algebra textbook written by Michael Atiyah and Ian G. Macdonald. fact_2 : Sir Michael Francis Atiyah ( ; born 22 April 1929) is an English mathematician specialising in geometry. Question: Introduction to Commutative Algebra is a well-known commutative algebra textbook written by Michael Francis Atiyah,who was born on which date, an English mathematician specialising in geometry?
Output (English): 22 April 1929