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The implication in the opposite direction is obvious. , to satisfy the formula a+ b= b + a . We shall now give a few further results in the same direction. 52. If a . p + b = b + a . v , O

A number of the preceding theorems remain valid if RC is replaced in them everywhere by RC'. 61(i),(ii). ,b,, ... different from 0 for which Among further properties of RC' the following should be mentioned: for every a E RC' the formula a+ b=a implies b = 0 and, more generally, the formula a+b=a+c implies b = c . 65. To conclude, we shall discuss briefly some algebraic generalizations of the results so far obtained. It may be noticed that, in the whole preceding discussion, when establishing properties of finite sums we did not use arbitrary sums of infinite sequences, but only special sums of this kind, in fact, the wth multiples.

For some f, (1) + c+d +c, + + (a+c+ d ) (c f ) = (b + c + d ) +c. 34, there is an e such that either (2) a+c+d+e=b+c+d and c+f=e+c or else (3) a + c + d= b +c + d + e and e + c + f = c. ( 2 ) directly implies the conclusion of (ii). 27 hence by (1) c+f=c, u+c+~+c=~+c+~+c and, by applying part (i) of our theorem (with p= l ) , we again arrive at the conclusion. Finally, (iii) is a simple consequence of (i). 44. o**
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