query: Show that if $r$ is nilpotent in a ring with identity, then $1-r$ is a unit in $R$
positive_passages.docid: 17895
positive_passages.text: Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$?
negative_passages.docid: 22563
negative_passages.text: Units and Nilpotents
negative_passages.docid: 34183
negative_passages.text: An equivalent condition with $\{0\}$ being the only nilpotent ideal
negative_passages.docid: 69649
negative_passages.text: If an ideal contains the multiplicative identity, then it is the whole ring
negative_passages.docid: 91180
negative_passages.text: Show $R \setminus S$ is a union of prime ideals
negative_passages.docid: 140437
negative_passages.text: If $a_0\in R$ is a unit, then  $\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$
negative_passages.docid: 81296
negative_passages.text: If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
negative_passages.docid: 39049
negative_passages.text: Ring With Unity As a Direct Sum of Non-Zero Ideals
negative_passages.docid: 131671
negative_passages.text: Let R be a ring such that $a^2 = a$ , $\forall a$ $\in R$ . Prove that R is commutative.
negative_passages.docid: 61526
negative_passages.text: An exercise about nilpotent quotient groups
negative_passages.docid: 32772
negative_passages.text: Proving units in a ring
negative_passages.docid: 103830
negative_passages.text: Each element of a ring is either a unit or a nilpotent element iff the ring has a unique prime ideal
negative_passages.docid: 18971
negative_passages.text: Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
negative_passages.docid: 4353
negative_passages.text: Unit group of power series ring
negative_passages.docid: 20613
negative_passages.text: Units (invertibles) of polynomial rings over a field
negative_passages.docid: 45310
negative_passages.text: If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.
negative_passages.docid: 38346
negative_passages.text: Commutative Ring: Nilpotent elements closed under addition?