I Understanding zero divisors & ##\mathbb{Z_m}## in Ring Theory

[chwala]

Am still looking at the literature, i need confirmation on $6$;

Now we know that;

My understanding- In reference to 6;
Let $r=3$ and $p=2$, then it follows that,
$3^2+3^2=18\mod2=0$
$(3+3)^2=36\mod2=0$

satisfies definition 2.3.1 (1)$3^2⋅ 3^2=81\mod2=1$
$(3⋅3)^2=81\mod2=1$

satisfies definition 2.3.1 (2)using another example say, $r=2$ and $p=3$, then it follows that,
$2^3+2^3=16\mod3=1$
$(2+2)^3=64\mod3=1$

satisfies definition 2.3.1 (1)
$2^3⋅ 2^3=64\mod3=1$
$(2⋅2)^3=64\mod3=1$

satisfies definition 2.3.1 (2)thus the pth power map is a ring homomorphism.

 

[chwala]

Also here; i need insight on the inverse of a matrix...

Are they assuming that in the given symmetric matrix; both $n$ values are units? What if we have non- unit $n$ values? ...then the inverse would not be equal to $n$.

 

[Maarten Havinga]

The p-power is a homomorphism indeed based on your observations. The $1^p=1$ part is trivial but when handing something in do add it.

As to the matrix inverse, there's a slight confusion. They mean the inverse of the homomorphism as a map, not the inverse of the matrix. It's a rather trivial statement.

 

[chwala]

The p-power is a homomorphism indeed based on your observations. The $1^p=1$ part is trivial but when handing something in do add it.

As to the matrix inverse, there's a slight confusion. They mean the inverse of the homomorphism as a map, not the inverse of the matrix. It's a rather trivial statement.

Thanks... we can't have $1^p=1$ because $p$ has to be prime ... unless I am missing something here.

 

[SammyS]

Also here; i need insight on the inverse of a matrix...

View attachment 301452Are they assuming that in the given symmetric matrix; both $n$ values are units? What if we have non- unit $n$ values? ...then the inverse would not be equal to $n$.

What do you mean by "both $n$ values" being "units" ?

As @Maarten Havinga states, they are referring to the inverse map .

Show that the map given as $\psi$ is the inverse (as in function inverse) of the map given as $\phi$ .

 

[chwala]

What do you mean by "both $n$ values" being "units" ?

As @Maarten Havinga states, they are referring to the inverse map .

Show that the map given as $\psi$ is the inverse (as in function inverse) of the map given as $\phi$ .

What do you mean by "both $n$ values" being "units" ? ...$n=1$.

Now understood.