  Let $\mathcal{A}$ be a weakly sequentially complete Banach algebra containing
a bounded approximate identity that is an ideal in its second dual
$\mathcal{A}^{\ast\ast}$, we call such an algebra a Wesebai algebra. In the
present paper we examine the Arens regularity properties of closed ideals of
algebras in the Wesebai class. We observe that, although Wesebai algebras are
always strongly Arens irregular, a variety of Arens regularity properties can
be observed within their closed ideals.
  After characterizing Arens regular ideals and strongly Arens irregular
ideals, we proceed to particularize to the main examples of \wasabi algebras,
the convolution group algebras $L^1(G)$, $G$ compact, and the Fourier algebras
$A(\Gamma)$, $\Gamma$ discrete and amenable. We find examples of Arens regular
ideals in $L^1(G)$ and $A(\Gamma)$, both reflexive and nonreflexive and
examples of strongly Arens irregular ideals that are not in the \wasabi class.
For this, we construct, in many noncommutative groups, a new class of Riesz
sets which are not $\Lambda(p)$, for any $p>1$. Our approach also shows that
every infinite Abelian group contains a Rosenthal set that is not $\Lambda(p)$,
for any $p>0$. These latter results could be of independent interest.
