Algebraic K-theory is an interesting invariant of rings and ring spectra which has connections to many mathematical fields including number theory, geometric topology, and algebraic geometry. While there is great interest in algebraic K-theory, it is difficult to compute. One successful approach is via trace methods. In this approach one utilizes trace maps from algebraic K-theory to more computable invariants which approximate algebraic K-theory. One of these trace maps is from algebraic K-theory to topological Hochschild homology (THH), which is an invariant of ring spectra. One of the main tools to compute THH is the B\"okstedt spectral sequence, and the algebraic structure in this spectral sequence facilitates computations.In recent years, several equivariant analogues of algebraic K-theory and THH have emerged. One such analogue is $C_n$-twisted THH, an invariant of ring $C_n$-spectra, which was defined by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell \cite{twTHH}. To compute twisted THH there is an equivariant B\"okstedt spectral sequence, constructed by Adamyk, Gerhardt, Hess, Klang, and Kong \cite{AGHKK}.This thesis explores the algebraic structures of twisted THH, and the equivariant B\"okstedt spectral sequence. Classically, if $A$ is a commutative ring spectrum, \cite{EKMM} and \cite{MSV} show that $\THH(A)$ is an $A$-Hopf algebra in the stable homotopy category. Angeltveit and Rognes extend this algebraic structure to the B\"okstedt spectral sequence and prove that under some conditions, the B\"okstedt spectral sequence is a spectral sequence of $H_*(A;\FF_p)$-Hopf algebras for $p$ prime \cite{Angeltveit-Rognes}. In this thesis we show that for $p$ prime and $R$ a commutative ring $C_p$-spectrum, $\THH_{C_p}(R)$ is an $R$-algebra in the $C_p$-equivariant stable homotopy category. Further, for $p \geq 5$ prime and $R$ a commutative ring $C_p$-spectrum, $\THH_{C_p}(R)$ is a non-counital $R$-bialgebra in the $C_p$-equivariant stable homotopy category. We also extend these results to the equivariant B\"okstedt spectral sequence, proving that under appropriate flatness conditions it is a spectral sequence of non-counital bialgebras.
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