factors with minus sign

The sign (cf. plus sign, opposite number) rule

$(+a)(-b)=-(ab),$(1)

derived in theparent entry (http://planetmath.org/productofnegativenumbers)and concerning numbers and elements $a$, $b$ of an arbitraryring, may be generalised to the following

Theorem.  If the sign of one factor (http://planetmath.org/Product) in a ring product is changed, the sign of the product changes.

Corollary 1.  The product of real numbers is equal to the product of their absolute values equipped with the “$-$” sign if the number of negative factors is odd and with “$+$” sign if it is even. Especially, any odd power of a negative real number is negative and any even power of it is positive.

Corollary 2.  Let us consider natural powers of a ring element. If one changes the sign of the base, then an odd power changes its sign but an even power remains unchanged:

$${(-a)}^{2n+1}=-a^{2n+1},{(-a)}^{2n}=a^{2n}\mathit{\hspace{1em}\hspace{1em}}(n\in \mathbb{N})$$