Research Space for Linear Algebra & Discrete Mathematics
2010-02-24
[離散] - 布林代數2
2.Suppose that(R,+,．)and(S,⊕,⊙) are two rings with zero element Zr and Zs respectively. Given a ring homorphism f:R→S,let K={aεR|f(a)=Zs} Prove that (a)(K,．,+) is a ring (b)K is an ideal of R
(a) 證 (K,+,．) 為 (R,+,．) 之 subring: ∀ a,b∈K, (i) f(a-b)=f(a)-f(b)=0∈K (ii) ∀ a,b∈K, f(ab)=f(a)⊙f(b)=0∈K 所以 (K,+,．) 為 ring (b) ∀ a∈K, b∈R, 因為 f(ab)=f(ba)=0, 所以 K is an ideal of R
(a) 證 (K,+,．) 為 (R,+,．) 之 subring: ∀ a,b∈K,
回覆刪除(i) f(a-b)=f(a)-f(b)=0∈K
(ii) ∀ a,b∈K, f(ab)=f(a)⊙f(b)=0∈K
所以 (K,+,．) 為 ring
(b) ∀ a∈K, b∈R, 因為 f(ab)=f(ba)=0,
所以 K is an ideal of R
謝謝助教
回覆刪除(i) f(a-b)=f(a)-f(b)=0∈K
回覆刪除^^^
這邊的 - 是不是要有圈圈，因為是右邊的運算
Prove that
回覆刪除(a)(K,．,+) is a ring
(b)K is an ideal of R
題目問"(K,．,+)" 乘法和加法是題目有打錯嗎?不然應該連"(K,．,+)" 都不是環了吧@@
Tse: 是的, 我想你應該知道所以我就懶的打了
回覆刪除pai: 他應該只是放反了