
48 Chapter 2 Ringoids and ve c toids
(h) a o D a C.o/ D
(D4)
a Co D
(D2)
a, i.e., o is a right neutral element of subtraction.
Assume o
0
is another one. Then for all a 2 R, ao
0
D aC.o
0
/ D a )
(a)
o
0
D o
)
(g)
o
0
D o.
(i) .a/=.b/ D ..e/a/=..e/b/ D
(D9)
..e/..e/a//=b D
(c)
a=b.
(j) .e/=.e/ D
(i)
e=e D
(D7)
e.
(k) Assume that e and e
0
are right neutral elements. Then e D e
0
=e
0
D e
0
.
(l) Assume that e D e.Thena b )a b in contradiction to (OD2).
(m) a b ^ c d ) a C c b C c b C d.
(n) a<b:, a b ^ a ¤ b )
(OD2)
b a ^b ¤a ,: b<a.
(o) o a b ^ o c d )
(OD3)
o ac bc bd.
(p) a b o ^ c d o
) o b a ^ o d c )
(o)
o .b/.d/