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My algebra book introduces sum, intersection and product of ideals (in non-commutative rings), and then says that all three operations are commutative and associative, without proof.

I see no reasons why the product of ideals should be commutative, but I wasn't able to find a counterexample either.

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    @Pete: Many times when thinking about a statement, I cannot see why it should be true, but cannot find a counter-example. Without proving it. When I ask people I give the clear distinction whether or not I was thinking about it or trying to prove it.2011-06-24

2 Answers 2

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It is not true that ideal product is commutative in non-commutative (associative, unital) rings. An easy example is given by triangular rings (see for instance pp 17-22, especially p. 18 of Lam's First Course in Non-commutative Rings).

You can easily construct a finite counterexample with 8 elements: Let a be the ring of upper-triangular matrices; let i be the (two-sided) ideal of a consisting of strictly upper triangular matrices, and let j be the (two-sided) ideal consisting of matrices whose second row is zero. Then ij = 0, but ji = i. This works over any field, in particular the field with 2 elements.

You can verify this in GAP:

gap> a:=AlgebraWithOne(GF(2),[[[0,1],[0,0]],[[1,0],[0,0]]]*One(GF(2)));  gap> i:=Ideal(a,[[[0,1],[0,0]]]*One(GF(2))); , (1 generators)> gap> j:=Ideal(a,[[[1,0],[0,0]],[[0,1],[0,0]]]*One(GF(2)));; gap> ij:=Ideal(a,Concatenation(List(i,x ->List(j,y->x*y))));; gap> ji:=Ideal(a,Concatenation(List(i,x ->List(j,y->y*x))));; gap> Dimension(ij);Dimension(ji); 0 1 gap> ji = i; true 
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    Your answer would have been nice even without the verification in GAP. But I installed GAP now and tried your code. Nice.2011-06-23
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For another example, consider the ring $\mathbb{Z}\langle x,y\rangle$ be the ring of polynomials with integer coefficients in two noncommuting variables $x$ and $y$ (this is the same as the integral semigroup ring of the free monoid of rank $2$). It's almost like the usual polynomial ring, except that the monomials are given by words in $x$ and $y$; so that, for example, the weight two monomials are $x^2$, $xy$, $yx$, and $y^2$, all distinct; the weight three monomials are $x^3$, $x^2y$, $xyx$, $yx^2$, $xy^2$, $yxy$, $y^2x$, and $y^3$, all distinct; etc.

Let $I = (x)$, the principal ideal generated by $x$. It consists of all elements of the form $\sum_{i=1}^n p_i(x,y)xq_i(x,y)$ with $p_i,q_i\in\mathbb{Z}\langle x,y\rangle$, $n\geq 0$; i.e., all polynomials in which every monomial has an $x$ in it, somewhere. Likewise, $J=(y)$ consists of all polynomials in which every monomial has a $y$ in it somewhere.

Now, clearly, $xy\in IJ$. But it cannot be in $JI$, because every monomial in every nonzero element of $JI$ must have an $x$ that follows its first $y$, which is not the case for $xy$.

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    Right, this is a "universal" counterexample.2011-06-23