21. Alternatively, a near-field is a near-ring in which there is a multiplicative identity, and every non-zero element has a multiplicative inverse. 22. Instead, the mapping is from the space to the zero element in ! ( just the ordinary zero 0 ): 23. They form a ring, that is, one can multiply, add and subtract them, but not always divide by a non-zero element . 24. Its standard basis, consisting of the sequences having only one non-zero element , which is equal to 1, is a countable Hamel basis. 25. In general the zero element of a ring is unique and typically denoted as 0 without any subscript indicating the parent ring. 26. To define them he uses diagonal matrices " A ij " having only + 1 or " 1 for non-zero elements . 27. Bulu?" et al . " present a sparse matrix data structure that Z-orders its non-zero elements to enable parallel matrix-vector multiplication. 28. One can always divide any non-zero element of a Hilbert space by its norm and obtain a " normalized " state vector. 29. In general the zero element of a ring is unique and typically denoted as 0 without any subscript to indicate the parent ring. 30. In a null set is the set with zero elements ; and in measure theory, a null set is a set with zero measure.
