Need help deciphering this number theory problem

[1MileCrash]

Homework Statement

What does triangle line mean? What is "+" for sets here?

Once I know that, if I need assistance, I will show an attempt. Otherwise I will be satisfied. :)

Homework Equations

The Attempt at a Solution

 

[micromass]

The triangle means "is ideal of". So $K_1$ and $K_2$ are ideals of $R$. We then define

K_1 + K_2 = \{a+b~\vert~a\in K_1,~b\in K_2\}

This is again an ideal of the ring.

 

[1MileCrash]

Ok, and while we're at it, what on Earth does equivalence mod "an ideal" mean? Does it mean the full set of equivalences for all elements in an ideal?

I am referring to the statement "x =K1 a." I just realized the picture is sideways.

And why is it stating that "if the following system is solvable.." since when is a system of congruences with coefficients of 1 ever not solvable?

Thank you.

 

[micromass]

Ok, and while we're at it, what on Earth does equivalence mod "an ideal" mean? Does it mean the full set of equivalences for all elements in an ideal?

I am referring to the statement "x =K1 a." I just realized the picture is sideways.

It means $x-a\in K_1$.

And why is it stating that "if the following system is solvable.." since when is a system of congruences with coefficients of 1 ever not solvable?

For example, the system $x\equiv_2 0$, $x\equiv_2 1$ is not solvable. This is equivalence modulo the ideal $2\mathbb{Z}$.

 

[1MileCrash]

Alrighty, thanks for filling in the gaps for me, hah!