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Prime Avoidance Lemma
In algebra, the prime avoidance lemma says that if an ideal ''I'' in a commutative ring ''R'' is contained in a union of finitely many prime ideals ''P''''i'''s, then it is contained in ''P''''i'' for some ''i''. There are many variations of the lemma (cf. Hochster); for example, if the ring ''R'' contains an infinite field or a finite field of sufficiently large cardinality, then the statement follows from a fact in linear algebra that a vector space over an infinite field or a finite field of large cardinality is not a finite union of its proper vector subspaces. Statement and proof The following statement and argument are perhaps the most standard. Statement: Let ''E'' be a subset of ''R'' that is an additive subgroup of ''R'' and is multiplicatively closed. Let I_1, I_2, \dots, I_n, n \ge 1 be ideals such that I_i are prime ideals for i \ge 3. If ''E'' is not contained in any of I_i's, then ''E'' is not contained in the union \cup I_i. Proof by induction on ''n'': The id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
