  We reobtain and often refine prior criteria due to Kaplansky, McGovern,
Roitman, Shchedryk, Wiegand, and Zabavsky--Bilavska and obtain new criteria for
a Hermite ring to be an \textsl{EDR}. We mention three criteria: (1) a Hermite
ring $R$ is an \textsl{EDR} iff for all pairs $(a,c)\in R^2$, the product
homomorphism $U(R/Rac)\times U\bigl(R/Rc(1-a)\bigr)\to U(R/Rc)$ between groups
of units is surjective; (2) a reduced Hermite ring is an \textsl{EDR} iff it is
a pre-Schreier ring and for each $a\in R$, every zero determinant unimodular
$2\times 2$ matrix with entries in $R/Ra$ lifts to a zero determinant matrix
with entries in $R$; (3) a B\'{e}zout domain $R$ is an \textsl{EDD} iff for all
triples $(a,b,c)\in R^3$ there exists a unimodular pair $(e,f)\in R^2$ such
that $(a,e)$ and $(be+af,1-a-bc)$ are unimodular pairs. We use these criteria
to show that each B\'{e}zout ring $R$ that is an $(SU)_2$ ring (as introduced
by Lorenzini) such that for each nonzero $a\in R$ there exists no nontrivial
self-dual projective $R/Ra$-module of rank $1$ generated by $2$ elements (e.g.,
all its elements are squares), is an \textsl{EDR}.
