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Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.
Some Related Sentences
Let and be
Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.
Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.
Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.
Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.
Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.
Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.
Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let and unital
Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).
Let C be the category of vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K ( assumed to be unital and associative ).
Let be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, U < sub > q </ sub >( G ), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators ( where λ is an element of the weight lattice, i. e. for all i ), and and ( for simple roots, ), subject to the following relations:
Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → U < sub > L </ sub >, ( notation as above ) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → A < sub > L </ sub > there exists a unique unital algebra homomorphism g: U → A such that: f (-) = g < sub > L </ sub > ( h (-)).
Let A and B be two commutative rings with unity, and let f: A → B be a ( unital ) ring homomorphism.
Let and commutative
Let g be a Lie algebra, h a maximal commutative Lie subalgebra consisting of semi-simple elements ( sometimes called Cartan subalgebra ) and let V be a finite dimensional representation of g. If g is semisimple, then g = g and so all weights on g are trivial.
Let P be a finitely generated projective module over a commutative ring R and X be the spectrum of R. The rank of P at a prime ideal in X is the rank of the free-module.
Let be a commutative ring with 1, e. g. ( Instead we can take to be a field and can replace by the field ).
Let A be a superalgebra over a commutative ring K. The submodule A < sub > 0 </ sub >, consisting of all even elements, is closed under multiplication and contains the identity of A and therefore forms a subalgebra of A, naturally called the even subalgebra.
Let R be a ( Noetherian, commutative ) regular local ring and P and Q be prime ideals of R. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra.
Let R be a commutative ring with prime characteristic p ( an integral domain of positive characteristic always has prime characteristic, for example ).
* Let R ⊂ S be an integral extension of commutative rings, and P a prime ideal of R. Then there is a prime ideal Q in S such that Q ∩ R = P. Moreover, Q can be chosen to contain any prime Q < sub > 1 </ sub > of S such that Q < sub > 1 </ sub > ∩ R ⊂ P.