Previous investigations on protein associations with diet quality and obesity still have inconclusive findings, possibly due to how protein intake was expressed. This study aimed to compare how different ways of expressing total protein intake may influence its relationships with diet quality and obesity. Usual protein intake was estimated from the 2011–12 Australian National Nutrition and Physical Activity Survey (n = 7637 adults, ≥19 years), expressed in grams (g/d), percent energy (%EI), and grams per actual kilogram body weight (g/kgBW/d). Diet quality was assessed using the 2013 Dietary Guidelines Index, and obesity measures included Body Mass Index (BMI) and waist circumference (WC). Sex-stratified multiple linear and logistic regressions were performed and adjusted for potential confounders. Total protein (g/d) was directly associated with diet quality (males, β = 0.15 (95% CI 0.12, 0.19); females, β = 0.25 (0.22, 0.29)), and this association was consistent across units. Protein intake (g/d) was directly associated with BMI (males, β = 0.07% (0.04%, 0.11%); females, β = 0.09% (0.04%, 0.15%)), and WC (males, β = 0.04 (0.01, 0.06); females, β = 0.05 (0.00, 0.09)). While in males, protein as %EI was associated with higher WC, no association was found in females. Adults with higher protein intake (g/d) had higher odds of overweight/obesity (males, OR = 1.01 (1.00, 1.01); females, OR = 1.01 (1.00, 1.01)), and central overweight/obesity (females, OR = 1.01 (1.00, 1.01)), but no significant association with females odds of overweight/obesity when protein was expressed in %EI. In conclusion, protein intake was positively associated with diet quality and obesity, yet these associations were stronger for women. The effect sizes also varied by measurement unit due to the different scales of those units.

We prove that a ring R is an $n \times n$ matrix ring (that is, $R \cong \mathbb {M}_n(S)$ for some ring S) if and only if there exists a (von Neumann) regular element x in R such that $l_R(x) = R{x^{n-1}}$. As applications, we prove some new results, strengthen some known results and provide easier proofs of other results. For instance, we prove that if a ring R has elements x and y such that $x^n = 0$, $Rx+Ry = R$ and $Ry \cap l_{R}(x^{n-1}) = 0$, then R is an $n \times n$ matrix ring. This improves a result of Fuchs [‘A characterisation result for matrix rings’, Bull. Aust. Math. Soc. 43 (1991), 265–267] where it is proved assuming further that the element y is nilpotent of index two and $x+y$ is a unit. For an ideal I of a ring R, we prove that the ring $(\begin {smallmatrix} R & I \\ R & R \end {smallmatrix})$ is a $2 \times 2$ matrix ring if and only if $R/I$ is so.

Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull. 48(1) (2005), 80–89].

Let $R$ be a ring. A map $f\,:\,R\,\to \,R$ is additive if $f(a\,+\,b)\,=\,f(a)\,+\,f(b)$ for all elements $a$ and $b$ of $R$ . Here, a map $f\,:\,R\,\to \,R$ is called unit-additive if $f(u\,+\,v)\,=\,f(u)\,+\,f(v)$ for all units $u$ and $v$ of $R$ . Motivated by a recent result of $\text{Xu}$ , $\text{Pei}$ and $\text{Yi}$ showing that, for any field $F$ , every unit-additive map of ${{\mathbb{M}}_{n}}(F)$ is additive for all $n\,\ge \,2$ , this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring $R$ is additive if and only if either $R$ has no homomorphic image isomorphic to ${{\mathbb{Z}}_{2}}\,\text{or}\,R/J(R)\,\cong \,{{\mathbb{Z}}_{2}}\,$ with $2\,=\,0$ in $R$ . Consequently, for any semilocal ring $R$ , every unit-additive map of ${{\mathbb{M}}_{n}}(R)$ is additive for all $n\,\ge \,2$ . These results are further extended to rings $R$ such that $R/J(R)$ is a direct product of exchange rings with primitive factors Artinian. A unit-additive map $f$ of a ring $R$ is called unithomomorphic if $f(uv)\,=\,f(u)f(v)$ for all units $u$ , $v$ of $R$ . As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

Let R be the ring of linear transformations of a right vector space over a division ring D. Three results are proved: (1) if |D|>4, then for any a∈R there exists a unit u of R such that a+u,a−u and a−u−1 are units of R; (2) if |D|>3 , then for any a∈R there exists a unit u of R such that both a+u and a−u−1 are units of R; (3) if |D|>2 , then for any a∈R there exists a unit u of R such that both a−u and a−u−1 are units of R. The second result extends the main result in H. Chen, [‘Decompositions of countable linear transformations’, Glasg. Math. J. (2010), doi:10.1017/S0017089510000121] and the third gives an affirmative answer to the question raised in the same paper.

Let R be a commutative ring with an automorphism ∞ of finite order n. An element f of the skew polynomial ring R[x, α] is nilpotent if and only if all coefficients of fn are nilpotent. (The case n = 1 is the well-known description of the nilpotent elements of the ordinary polynomial ring R[x].) A characterization of the units in R[x, α] is also given.