How to prove this basic fact: ( a ) ( − 1 ) = − a. How to prove that an additive inverse can be expressed multiplicatively?

Question

How to prove this basic fact:   (  a  )  (  −  1  )  =  −  a. How to prove that an additive inverse can be expressed multiplicatively?

Angelo Murray · Accepted Answer

Let   R be a commutative ring and let   a  ,  b  ∈  R. Let   0 be the zero element of   R and   −  a be the additive inverse of   a. Here are some basic facts:(1)   0  a  =  0.Proof:   0  +  0  =  0. Multiplying with a gives   a  0  =  a  (  0  +  0  )  =  a  0  +  a  0. Adding   −  a  0 on both sides gives   0  =  a  0.(2)   −  (  −  a  )  =  a.Proof:   −  (  −  a  ) and   a are both additive inverses of   −  a. By the uniqueness of add. inverses,   −  (  −  a  )  =  a.(3)   (  −  a  )  b  =  a  (  −  b  )  =  −  a  b.Proof:   a  b  +  (  −  a  )  b  =  (  a  +  (  −  a  )  )  b  =  0  b  =  0 and so   (  −  a  )  b is inverse to   a  b. By the uniqueness of add. inverses,   (  −  a  )  b  =  −  a  b.(4)   (  −  a  )  (  −  b  )  =  a  bProof:   (  −  a  )  (  −  b  )  =  −  (  (  −  a  )  b  )  =  −  (  −  (  a  b  )  ) by (3) and   −  (  −  a  b  )  =  a  b by (2).In particular, by (3),   a  (  −  1  )  =  −  (  a  1  )  =  −  a.

skylsn · Accepted Answer

You can use the fact the                    a        +        (        −        1        )        a                            =        1        ×        a        +        (        −        1        )        ×        a                                          =                          (                1        +        (        −        1        )                          )                a                                          =        0        ×        a                                          =        0.            Of course, I am using here the fact that   0  ×  a  =  0, which can be proved as follows:                    0                            =        0        ×        a        −        0        ×        a                                          =        (        0        +        0        )        ×        a        −        0        ×        a                                          =        (        0        ×        a        +        0        ×        a        )        −        0        ×        a                                          =        0        ×        a        +        (        0        ×        a        −        0        ×        a        )                                          =        0        ×        a        +        0                                          =        0        ×        a

How to prove this basic fact: (

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