I have difficulties with a task in linear algebra.
R is a ring and R = {a, b, c, d}
These are the tables for + and . in R:
+| a b c d .| a b c d
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a| a b c d a| a a a a
b| b a d c b| a b c d
c| c d a b c| a c c a
d| d c b a d| a d a d
I know the definition for ideal and I understand why for example we take c and we look at the . table for the row and column of the element c to find out which elements stand there - because a.x and x.a should stay in the ideal.
But I don't quite understand the role of the + table here. We have that a-x should be in the ideal, too. So what? When we see that multiplying c to whatever element we get a or c, we should look if c+a and a+c gives us a or c, too? I can't quite get this.
Can you please help me? Thanks very much in advance!
The addition table tells us that $a$ is the additive identity, i.e. $a=0$. The multiplication table tells us that $b$ is the multiplicative identity, i.e. $b=1$ so $R$ is a commutative ring with identity and we immediately get that $R$ and $(a)$ are ideals.
What are the non-trivial additive subgroups? We notice that $c+c=a=0$ and $d+d=a=0$ so $\{a,c\}$ and $\{a,d\}$ are additive subgroups. Are they ideals? A quick look at the multiplication table shows they are.
Are there any others? Since $b=1$ we know any ideal containing $b$ is the entire ring and since $c+d=b$ we know any any ideal containing both $c$ and $d$ must be the entire ring.
Therefore, the ideals are $R$, $(a)$, $(c)$ and $(d)$.