Zero Element
The identity element of an additive monoid or group or of any other algebraic structure (e.g., ring, module, abstract vector space, algebra) equipped with an addition. It is also called the additive identity and is denoted 0. The name and the symbol are borrowed from the ring of integers whose additive identity is, of course, number 0.
The zero element of a ring R has the property that a·0=0·a=0 for all a in R and, moreover, for every element x of an R-module M it holds that 0_R·x=0_M. Here, the indices distinguish the zero element of the ring from the zero element of the module. The latter also fulfils the rule a·0_M=0_M for all a in R.
The notation 0 is sometimes also used for the universal bound emptyset of a Boolean algebra A. In fact it behaves with respect to the operation ^ like a zero element with respect to multiplication, since a ^ emptyset=emptyset ^ a=emptyset for all a in A.
See also
Additive IdentityThis entry contributed by Margherita Barile
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Barile, Margherita. "Zero Element." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ZeroElement.html