query: If a unit and nilpotent element commute in a ring, their sum is also a unit.
positive_passages.docid: 22563
positive_passages.text: Units and Nilpotents
negative_passages.docid: 87946
negative_passages.text: Proving that the unities of a ring form a group under multiplication
negative_passages.docid: 144082
negative_passages.text: Every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix.
negative_passages.docid: 61391
negative_passages.text: Which finite groups are the group of units of some ring?
negative_passages.docid: 134622
negative_passages.text: Sum of nil right ideals
negative_passages.docid: 83115
negative_passages.text: A set of elements in a reduced unity ring
negative_passages.docid: 15904
negative_passages.text: Prime ideal and nilpotent elements
negative_passages.docid: 90239
negative_passages.text: $\mathbb{Z} \times \mathbb{Z}$ is cyclic.
negative_passages.docid: 45310
negative_passages.text: If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.
negative_passages.docid: 39049
negative_passages.text: Ring With Unity As a Direct Sum of Non-Zero Ideals
negative_passages.docid: 74139
negative_passages.text: Subrings of integral domains have the same identity element.
negative_passages.docid: 141149
negative_passages.text: Proof some module is direct sum by its submodules.
negative_passages.docid: 24259
negative_passages.text: Quotient ring of $T/I$
negative_passages.docid: 17895
negative_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: 121465
negative_passages.text: $R[x]$ has a subring isomorphic to $R$.
negative_passages.docid: 158873
negative_passages.text: Is every group the unit group of some ring?
negative_passages.docid: 156878
negative_passages.text: If every commutator is idempotent, then the ring is commutative