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A zero element (or an absorbing/annihilating element) is an element z such that for all s in S, z • s = s • z = z. This notion can be refined to the notions of left zero, where one requires only that z • s = z, and right zero, where s • z = z. [2]
In mathematics, the zero module is the module consisting of only the additive identity for the module's addition function. In the integers, this identity is zero, which gives the name zero module. That the zero module is in fact a module is simple to show; it is closed under addition and multiplication trivially.
The role of 0 as additive identity generalizes beyond elementary algebra. In abstract algebra, 0 is commonly used to denote a zero element, which is the identity element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined). (Such elements may also be called zero elements.)
The value ζ(0) = −1/2 is not determined by the functional equation, but is the limiting value of ζ(s) as s approaches zero. The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all nontrivial zeros lie in the critical strip where s has real part between 0 and 1.
It follows that the solutions of such an equation are exactly the zeros of the function . In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.
Since a ψ, ψ pair is a commuting element of the Grassmann algebra, it does not matter what order the pairs are in. If more than one ψ, ψ pair have the same k, the integral is zero, and it is easy to check that the sum over pairings gives zero in this case (there are always an even number of them). This is the Grassmann analog of the higher ...
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. [1] [2] For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings.
In a vector space, the null vector is the neutral element of vector addition; depending on the context, a null vector may also be a vector mapped to some null by a function under consideration (such as a quadratic form coming with the vector space, see null vector, a linear mapping given as matrix product or dot product, [4] a seminorm in a ...