In this work, all rings are commutative rings with identity. Let $R$ be a ring and $J(R)$ denote the Jacobson radical of $R$. A sequence $(a_1, a_2, \ldots , a_s, a_{s+1})$, $s \ge 1$, of elements in $R$ is said to be unimodular if 1 is in the ideal $(a_1, a_2, \ldots , a_s, a_{s+1} )$. A ring $R$ is said to be a $B$-ring, whenever for any unimodular sequence $(a_1, a_2, \ldots , a_s, a_{s+1})$, $s \ge 2$, of elements in $R$ with $(a_1, a_2, \ldots , a_{s-1} ) \not\subset J(R)$, there exists an element $b$ in $R$ such that $(a_1, a_2, \ldots , a_s + ba_{s+1} ) = R$. Besides some other results, two basic facts about $B$-rings will be proved and they are: 1) $R$ is a $B$-ring if and only if for any unimodular sequence $(a_1, a_2, a_3)$ of elements in $R$ with $a_1$ not in $J(R)$, there exists an element $b$ in $R$ such that $(a_1, a_2 + ba_3 ) = R$, and 2) A homomorphic image of a $B$-ring is again a $B$-ring. At the end, some applications of these results will be discussed.

Let $S = \{z=x+yj|x$ and $y$ real, $-\pi \lt y \leq \pi \}$ and let $S^* = S \cup \{ - \infty \}$. If $z = x_1 + y_1j$, $w = x_2 + y_2j \in S$, then define a cylindrical addition, $\oplus$, on $S^*$ by $z \oplus w = (x_1 + x_2) + [(y_1 + y_2) \pmod{2\pi}]j$ and $z \oplus - \infty = - \infty$ for all $z \in S^*$. Define a subaddition, $\odot$, on $S^*$ by $z \odot w = \log(e^z + e^w)$. Then $(C, +, \cdot) \cong (S^*, \oplus, \odot)$.

Subaddition has an application in signal processing.

A channel's fading can be modeled as the product of a slowly varying component and the transmitted signal. An amplitude-modulated signal can also be represented by a product of a carrier signal and envelope function.

The logarithmic function will transform a system modeled on a product to a conventional linear system that will yield to a classical attack. It is shown here that the logarithm, as a generalized superposition, will also transform a conventional linear system into another linear system, and therefore, nothing need be known about the original system before applying a logarithmic transformation.

This study investigates the detection of a changepoint due to location or scale or both location and scale changes in a sequence of univariate random variables. Two changepoint models, namely, the single change and the continuous change are considered and the appropriate test statistics are derived. The proposed test statistics are functions of two linear rank statistics, of which one is odd-translation invariant, sensitive to location shift and the other one is even-translation invariant, sensitive to scale shift. The asymptotic distributions of the proposed statistics are obtained under the null hypothesis.