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- Jun 22, 2012

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**Products of ideals of K[x1, x2, x3, x3]**

I am reading R.Y. Sharpe: Steps in Commutative Algebra. In chapter 2 on Ideals, on page 28 we find Exercise 2.27 which reads as follows: (see attachment)

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2.27 Exercise: Let \(\displaystyle K\) be a field, and let \(\displaystyle R = K[x_1, x_2, x_3, x_4] \), the ring of polynomials over K in indeterminates x_1, x_2, x_3, x_4.

Set \(\displaystyle I = Rx_1 + Rx_2 \) and \(\displaystyle J = Rx_3 + Rx_4 \)

Show that \(\displaystyle IJ \ne \{fg: \ f \in I, g \in J \} \)

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My problem is I seemed to have ended up showing that \(\displaystyle IJ = \{fg: \ f \in I, g \in J \} \) ... so obviously something is wrong with my working ...

**Can someone please explain my error(s)?**

My working is as follows:

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\(\displaystyle Rx_1 = \{ fx_1 \ | \ f \in R \} \)

and Rx_2, Rx_3, Rx_4 are defined similarly.

\(\displaystyle I = Rx_1 + Rx_2 \)

\(\displaystyle = \{ h+k \ | \ h \in Rx_1 , k \in Rx_2 \} \)

\(\displaystyle = \{ fx_1 + gx_2 \ | \ f, g \in R \} \)

and similarly

\(\displaystyle J = \{ hx_1 + kx_2 \ | \ h, k \in R \} \)

Then \(\displaystyle IJ = \) set of all finite sums of elements of the form \(\displaystyle lm \) with \(\displaystyle l \in I, m \in J \)

\(\displaystyle = \{ {\sum}_{i=1}^{n} l_im_i \ | \ n \in \mathbb{N}, l_i \in I, m_i \in J \} \)

\(\displaystyle = \{ {\sum}_{i=1}^{n} (f_ix_1 + g_ix_2)(h_ix_3 + k_ix_4) \ | \ f_i, g_i, h_i, k_i \in R \} \)

\(\displaystyle = \{ {\sum}_{i=1}^{n} f_ih_ix_1x_3 + f_ik_ix_1x_4 + g_ih_ix_2x_3 + g_ik_ix_2x_4 \ | \ f_i, g_i, h_i, k_i \in R \} \)

\(\displaystyle = \{ {\sum}_{i=1}^{n} l_ix_1x_3 + m_ix_1x_4 + p _ix_2x_3 + q_ix_2x_4 \ | \ l_i, m_i, p_i, q_i \in R \} \)

\(\displaystyle = lx_1x_3 + mx_1x_4 + px_2x_3 + x_2x_4 \ | \ l, m, p, q \in R \}\)

since we can put \(\displaystyle l_1 + l_2 + ... \ ... l_n = l \) and similarly with \(\displaystyle m, p, q \)

Now consider the set \(\displaystyle \{ fg: \ f\in I, g \in J \} \)

\(\displaystyle \{ fg: \ f\in I, g \in J \} \)

\(\displaystyle = \{ (l_1x_1 + m_1x_2)(p_1x_3 + q_1x_4) \ | \ l_1, m_1, p_1, q_1 \in R \} \)

\(\displaystyle = \{ l_1p_1x_1x_3 + l_1q_1x_1x_4 + m_1p_1x_2x_3 + m_1q_1x_2x_4 \ | \ l_1, m_1, p_1, q_1 \in R \} \)

\(\displaystyle = \{ lx_1x_3 + mx_1x_4 + px_2x_3 + qx_2x_4 \ | \ l, m, p, q \in R \} \)

BUT then \(\displaystyle IJ = \{fg: \ f \in I, g \in J \} \) ???

Can someone please explain my error(s)

Peter

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