Question

prove 3 and 4..؟

One خميس 26 مارس ۲۶۶۲ ص TENSOR HOMOMORPHISMS.pdf LEMMA: If MR MR RNR N' and fi 12,5 € (MM) 81,82,8 (N,N), then (1) + 12g = g)+2 g ), (2), (8, +82)=( g)+( 82) (3) / Ⓡ0=0®g=0, (4) IⓇINION PROOF: (1) The map (m,n) (m) + /2(m)) g() is R-balanced map MXN M' N', hence defines a unique Z map M N _Uit 128gM®N" given by m® (m)+2(m)) g(n). But from the relations of the tensor product we have [ +/ 2 g (m n)) =((m)+ f2(m)) g(n)- (m) g(71)+ /(m) g(n) - g)+(12 g)]( m n )] (2) Is quite similar to (1). (3) and (4) are easy exercise. If Picommutative rinn then werd both and M o re

Answer 1

To show for o@g=0 f ® 0 = f © (o+o) = f @o + foo (By property f@o - f® o + fo ) since MR é is an R-mochell (hence no anelian game alss) and f or MR @ Me - 140 c) < Me® Mé =) By Cancellation property of groups, we get, Similarly oxg=of Crete To Shows MON ® To show IM BIN IM® Recall : If firar and g: www me tro hommaplusme Then tog : vow row is a tensor homomophism, defined in (fog) u ( ow) := flu) g(w)

defined as Tow. In - MON IN : MON (IM IN) (man) = IM (m) IN (M) - mon & I MON (mon) in. (IM ® IN ) (man) = IMON (m@n) (': IM ® IN = IMON MON = generated by mon MEM, MEN)