Let R be an arbitrary ring and (n属于Z+) .If the set Sn is def
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Let R be an arbitrary ring and (n属于Z+) .If the set Sn is defined by
Sn = {(a 属于 R) l (n^k) *a = 0 for some k > 0}
determine whether Sn is a subring of R.
Sn = {(a 属于 R) l (n^k) *a = 0 for some k > 0}
determine whether Sn is a subring of R.
![Let R be an arbitrary ring and (n属于Z+) .If the set Sn is def](/uploads/image/z/2182900-4-0.jpg?t=Let+R+be+an+arbitrary+ring+and+%EF%BC%88n%E5%B1%9E%E4%BA%8EZ%2B%EF%BC%89+.If+the+set+Sn+is+def)
答案:正确
∀a,b∈Sn,∃k1,k2>0,(n^k1) *a = 0,(n^k2)*b= 0,令k=max(k1,k2),则 (n^k) *a = 0,(n^k) *b= 0,于是(n^k) *(a-b)= 0,
故 a-b∈Sn,故Sn is a subring of R..
∀a,b∈Sn,∃k1,k2>0,(n^k1) *a = 0,(n^k2)*b= 0,令k=max(k1,k2),则 (n^k) *a = 0,(n^k) *b= 0,于是(n^k) *(a-b)= 0,
故 a-b∈Sn,故Sn is a subring of R..
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