Further Development of the Theory of Arithmetics of Algebras

The writer recently* gave a new conception of integral elements of a rational associative algebra A having a modulus 1, which avoids the serious objections against all earlier conceptions. The integral elements of A are defined to be the elements which belong to a set 5 of elements having the following four properties: C (closure) : The sum, difference and product of any two elements of 5 are also elements of S. R (rank equationf) : For every element of S, the coefficients of the rank equation are all ordinary integers. U (unity): The set contains the modulus 1. M (maximal) : The set is a maximal (i.e., is not contained in a larger set having properties C, R, U). It is proved in §2 for the first time that there exists a set of integral elements in any rational algebra. The above conception of integral elements may be extended to algebras over an algebraic field (or any field for which the notion of integer is defined). In particular, quaternions over any quadratic field are investigated in §§ 4-9.