[Introduction to Commutative Algebra] Exercise 1.1

作者：Atiyah & Macdonald

複習一下交換代數，宅在家太無聊。但也因為心情愉快，較能專心算數學。雖然自己不善解人意，容易惹她生氣，她也真的生氣，但她也夠包容了，不去計較我的愚蠢。每次低潮都是向上的勇氣，如果做了後悔的事，那就不要繼續後悔。「這裡什麼都沒有，就算你再害羞、再沒自信，也沒有地方躲，現在你們只要相信曾經流過的汗水。」

Exercise 1.1: Let x be a nilpotent element of a ring A. Show that 1+x is a unit of A. Deduce that the sum of a nilpotent element and a unit is a unit.

Proof: x^n = 0 for some n > 0. So (-x)^n = 0 for that n. It means 1 - (-x)^n = (1 - (-x))(1 + (-x)^2 + ... + (-x)^{n-1}) = (1 + x)(1 + (-x)^2 + ... + (-x)^{n-1}) = 1. That is, 1 + x is a unit of A.

Next, all we need to do is to prove u+x is a unit where u is a unit and x is a nilpotent. It is trivial since u+x = u(1+u^{-1}x). u^{-1}x is nilpotent. Also, a unit multiplies by a unit is also a unit. ((uv)(v^{-1}u^{-1}) = 1.)

很多時候，常常需要回到原點。數學已經離我很遠，但那曾經的感動，還是耐人懷念。至於人間「溝通」兩三事，又是另種藝術值得學習與玩味。兩個字「溝通」，也要流過汗水。