nonsmi
lujvo x1 is the zero(like)/additive identity of structure/ring x2; often is denoted by ' 0_R ' (for structure/ring R, specified by x2) or by '0' when context is obvious Definition and rules may be specified in the second terbri; this definition does not suppose that the ring is not the 0ring (the trivial ring) with the mapping of all multiplications to 0 (in which case, the additive identity is also the multiplicative identity). The usage of "additive" and "multiplicative" in this context are defined by the ring. See also: pavysmi


nonsmitenfa
lujvo x1=t2 is an element in the set that underlies structure/ring x2≈s3 that is nilpotent in that structure with nilpotency x3=t3 (nonnegative integer according to the typical rules) x3 is the minimum positive exponent such that x1 multiplied by itself that many times (according to the definition of multiplication imposed by and endowing structure x2) is identically the zero(like) element in that structure; any greater power will likewise be zero(like). The zero(like) element is itself trivially nilpotent with nilpotency 1. Warning: This word is for nilpotent elements. Nilpotent groups, for example, should not be referred to by this word except when considered as whole objects that participate as elements in some larger structure. See also: nonsmi


pavysmi
lujvo x1 is the one(like) element/multiplicative identity of structure/ring x2; often is denoted by ' 1_R ' or ' I_R ' or by (when context is obvious) '1' or 'I', for structure/ring R (given by x2). Definition and rules may be specified in the second terbri; this definition does not suppose that the ring is not the 0ring (the trivial ring) with the mapping of all multiplications to 0 (in which case, the additive identity is also the multiplicative identity). The usage of "additive" and "multiplicative" in this context are defined by the ring. The n×n identity matrix over a given ring is an example of one such element. See also: nonsmi
