24

Jack Palmer Sanders

M is a ring spectrum if there are a pairing w: (M,M) - - M and a map

of spectra U: S -*M such that the following diagrams are homotopy

commutative for all p, q.

1

A

u

p + q

W A 1

M A M A M — ^ z S

p q r

M A M

p + q r

1 A W

q , r

p + q , r

p . q + r

M A M ^-^ M

p q + r p + q + r

M A M w

p q _ p , q

p + q

M A M w

q p q / P

Here T is the usual switch map. If M is a ring spectrum, then TT^ (M)

is an anticommutative ring with unit.

N is a module over the ring spectrum M if there is a pairing

g: (M,N) -*N such that

U

A

1

S AN — ~ • M A N

p q p q

P/q

P/q

p+q

homotopy commutes for all p, q. If N is a module over M, then TT^ (N)

is a left IT (M) -module.